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\title[Cyclic quasi-symmetric functions]
{Cyclic quasi-symmetric functions}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}
\author{%
	Ron M.\ Adin%
	\thanks{\href{mailto:radin@math.biu.ac.il}{radin@math.biu.ac.il}. Partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no.\ 1970/18.}%
	\addressmark{1}, 
	Ira M.\ Gessel%
	\thanks{\href{mailto:gessel@brandeis.edu}{gessel@brandeis.edu}. Partially supported by Simons Foundation Grant \#427060.}%
	\addressmark{2}, 
	Victor Reiner%
	\thanks{\href{mailto:reiner@math.umn.edu}{reiner@math.umn.edu}. Partially supported by NSF grant DMS-1601961.}%
	\addressmark{3}, \and
	Yuval Roichman%
	\thanks{\href{mailto:yuvalr@math.biu.ac.il}{yuvalr@math.biu.ac.il}. Partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no.\ 1970/18.}%
	\addressmark{1}%
}

%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}
\address{%
	\addressmark{1}%
	Department of Mathematics, Bar-Ilan University, 
	Ramat-Gan 52900, Israel \\
	\addressmark{2}%
	Department of Mathematics, Brandeis University,
	Waltham, MA 02453, USA \\
	\addressmark{3}%
	School of Mathematics, University of Minnesota,
	Minneapolis, MN 55455, USA% \\
	%\addressmark{4}%
	%Department of Mathematics, Bar-Ilan University, 
	%Ramat-Gan 52900, Israel%
}
%% put the date of submission here
%\date{October 28, 2018}
\received{November 15, 2018}

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\abstract{%
	The ring of cyclic quasi-symmetric functions is
	%and its combinatorial subring are 
	introduced in this paper. 
	A natural basis consists of
	fundamental cyclic quasi-symmetric functions; 
	%for the combinatorial subring 
	they arise as toric $P$-partition enumerators, for toric posets $P$ with a total cyclic order.
	The associated structure constants are determined by cyclic shuffles of permutations. 
	%We then prove the following positivity phenomenon: 
	For every non-hook shape $\lambda$, the coefficients in the expansion of the Schur function $s_\lambda$ in terms of fundamental cyclic quasi-symmetric functions are nonnegative. 
	%The proof relies on the existence of a cyclic descent map on the standard Young tableaux (SYT) of shape $\lambda$. 
	%%Combining this setting a ring homomorphism to formal power series,
	The theory has applications to  
	%the distribution of cyclic descents on 
	the enumeration of cyclic shuffles and SYT
	%of cyclic
	%circular permutations %with prescribed 
	by cyclic descents. 
	%We conclude by providing a cyclic analogue of Solomon's descent algebra.
}

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%\keywords{quasi-symmetric function, toric poset, cyclic shuffle, cyclic descent}

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	\marginpar{\textsc{ToDo}} \framebox{\begin{minipage}[c]{0.95
				\textwidth}
			#1 \end{minipage}}\vspace{5 mm}\par}


\newcommand{\NN}{{\mathbb {N}}}
\newcommand{\CC}{{\mathbb {C}}}
\newcommand{\QQ}{{\mathbb {Q}}}
\newcommand{\RR}{{\mathbb {R}}}
\newcommand{\ZZ}{{\mathbb {Z}}}

\newcommand{\then}{\Rightarrow}

\newcommand{\spn}{{\operatorname{span}}}
\newcommand{\Aut}{{\operatorname{Aut^\cyc}}}
\newcommand{\Des}{{\operatorname{Des}}}
\newcommand{\cDes}{{\operatorname{cDes}}}
\newcommand{\cyc}{{\operatorname{cyc}}}
\newcommand{\cdes}{{\operatorname{cdes}}}
\newcommand{\des}{{\operatorname{des}}}
\newcommand{\maj}{{\operatorname{maj}}}
\newcommand{\ch}{{\operatorname{ch}}}
\newcommand{\shape}{{\operatorname{shape}}}
\newcommand{\SYT}{{\operatorname{SYT}}}
\newcommand{\bded}{{\operatorname{bd}}}
\newcommand{\co}{{\operatorname{co}_n}}
\newcommand{\cc}{{\operatorname{cc}_n}}

\newcommand{\Comp}{{\operatorname{Comp}}}
\newcommand{\cComp}{{\operatorname{cComp}}}
\newcommand{\cnes}{{c2_0^{[n]}}}
\newcommand{\ccon}{{\operatorname{cyc-comp}(n)}}

\DeclareMathOperator{\QSym}{QSym}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\cQSym}{cQSym}

\DeclareMathOperator{\Sh}{Sh}

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\newcommand{\cs}{{\tilde{s}}}
\newcommand{\hM}{{\widehat{M}}}
\newcommand{\hF}{{\widehat{F}}}

\newcommand{\aaa}{{\mathbf{a}}}
\newcommand{\xx}{{\mathbf{x}}}
\newcommand{\ttt}{{\mathbf{t}}}
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\newcommand{\AAA}{{\mathcal{A}}}
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\newcommand{\DAG}[1]{\accentset{\rightharpoonup}{#1}}


\newcommand{\SteinbergTorus}{{\widetilde{\Delta}}}

\newcommand{\wcDes}{\cDes_*}%{\widehat{\cDes}}
\newcommand{\wcdes}{\cdes_*}%{\widehat{\cDes}}

\newcommand{\AffDes}{{\operatorname{cDes}}}
%\newcommand{\AffDes}{{operatorname{AffDes}}}
%\newcommand{\AffDes}{\widetilde{{operatorname{Des}}}}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
	
	\maketitle
	
	%\tableofcontents
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section{Introduction}
	
	%Recall that
	The graded rings $\Sym$ %(sometimes denoted $\Lambda$)
	and $\QSym$,
	of symmetric and quasi-symmetric functions, respectively,
	%,  which satisfy $\Sym \subsetneq \QSym$, 
	have many applications to enumerative combinatorics, as well as to other branches of mathematics; see, e.g., %\cite{Gessel} 
	\cite[Ch.\ 7]{EC2}.
	This paper introduces two intermediate objects: the graded ring $\cQSym$ of cyclic quasi-symmetric functions, 
	and its %combinatorial 
	subring $\cQSym^-$.
	
	\smallskip
	
	The rings $\Sym$, $\QSym$ and $\cQSym$ may be defined via invariance properties. 
	A formal power series $f\in \ZZ[[x_1,x_2,\ldots]]$ of bounded degree is {\em symmetric} if 
	for any $t \ge 1$, any two sequences $i_1, \ldots, i_t$
	and $j_1, \dots, j_t$ of distinct positive integers (indices), 
	and any sequence $m_1, \ldots, m_t$ of positive integers (exponents), 
	the coefficients of $x_{i_1}^{m_1} \cdots x_{i_t}^{m_t}$ and 
	$x_{j_1}^{m_1} \cdots x_{j_t}^{m_t}$ in $f$ are equal. 
	We call $f$ {\em quasi-symmetric} if 
	for any $t \ge 1$, any two {\em increasing} sequences $i_1 < \dots < i_t$ and $j_1 < \dots < j_t$ of positive integers, 
	and any sequence $m_1, \ldots, m_t$ of positive integers, 
	the coefficients of $x_{i_1}^{m_1} \cdots x_{i_t}^{m_t}$ and 
	$x_{j_1}^{m_1} \cdots x_{j_t}^{m_t}$ in $f$ are equal. 
	%
	%\smallskip
	%
	%A formal power series $f\in \ZZ[[x_1,x_2,\dots]]$ of bounded degree is 
	
	
	\begin{defn}\label{def:cQSym}
		A {\em cyclic quasi-symmetric function} is
		a formal power series $f \in \ZZ[[x_1, x_2, \ldots]]$ of bounded degree such that,
		for any $t \ge 1$, any two increasing sequences $i_1 < \dots < i_t$
		and $i'_1 < \dots < i'_t$ of positive integers, any sequence
		$m = (m_1, \ldots, m_t)$ of positive integers, and any {\em cyclic shift}
		$m' = (m'_1, \ldots, m'_t)$ of $m$, the coefficients of
		$x_{i_1}^{m_1} \cdots x_{i_t}^{m_t}$ and 
		$x_{i'_1}^{m'_1} \cdots x_{i'_t}^{m'_t}$ in $f$ are equal. 
	\end{defn}
	
	Denote by $\cQSym$ the set of all cyclic quasi-symmetric functions, 
	and by $\cQSym_n$ the set of all cyclic quasi-symmetric functions
	which are homogeneous of degree $n$.
	%\begin{obs}\label{t:QSym_subring}
	%$\cQSym$ is an abelian groups.
	It will be shown that $\cQSym$ is  a graded ring;
	see \cref{t:cQSym_is_a_ring}.
	
	\smallskip
	
	Toric posets were recently introduced by Develin, Macauley and Reiner~\cite{DMR}. 
	A toric analogue of $P$-partitions is presented in \cref{sec:toric}.
	Toric $P$-partition enumerators, in the special case of total cyclic orders, 
	form a convenient $\QQ$-basis for a ring $\cQSym^-$, which is a subring of $\cQSym$.
	%the ring of cyclic quasi-symmetric functions.
	A slightly extended set actually forms a $\QQ$-basis for $\cQSym$ itself.
	The elements of this basis are called 
	{\em fundamental cyclic quasi-symmetric functions},
	are indexed by cyclic compositions of a positive integer $n$
	(equivalently, by cyclic equivalence classes of nonempty subsets $J \subseteq [n]$), 
	and are denoted $F^\cyc_{n,[J]}$.
	Normalized versions of them actually form $\ZZ$-bases for $\cQSym$ and $\cQSym^-$; 
	see %Lemma~\ref{t:Fcyc_basis} and Observation
 \cref{t:proper_basis}.
	%The ring $\cQSym$ may be nicknamed ``algebraic'' (containing,
	%%all cyclic quasi-symmetric functions, and 
	%in particular, all symmetric functions), 
	%while its subring $\cQSym^-$ may be described as ``combinatorial'' %(containing only basis elements corresponding to actual cyclic descent sets of permutations).
	%(forming a natural context when dealing with cyclic descent sets of combinatorial objects).
	
	%\medskip
	
	A toric analogue of Stanley's fundamental decomposition lemma for $P$-partitions~\cite[Lemma 3.15.3]{EC1}, given in \cref{cyclic-fundamental-lemma} below, 
	%The special case of total cyclic order 
	is applied to provide a combinatorial interpretation of the resulting structure constants in terms of shuffles of cyclic permutations (more accurately, cyclic words), as follows.
	
	%\todo{Define cyclic permutations and cyclic shuffles}
	
	%\smallskip
	
	For a finite set $A$ of size $a$, let $\symm_A$ be 
	the set of all bijections $u \colon [a] \to A$,
	viewed as words $u = (u_1, \ldots, u_a)$. 
	Elements of $\symm_A$ will be called {\em bijective words}, or simply {\em words}.
	%If $A = [a]$ then $\symm_A$ is the symmetric group $\symm_a$, whose elements are genuine permutations.
	If $A$ is a finite set of integers, or any finite totally ordered set, define the {\em cyclic descent set} of $u \in \symm_A$  by
	%was first defined by Cellini~\cite{Cellini}  as
	\begin{equation}
	\label{e.cellini}
	\cDes(u) := \{1 \leq i \leq a \,:\, u_i > u_{i+1} \}
	\subseteq [a],
	\end{equation}
	with the convention $u_{a+1} := u_1$.
	The {\em cyclic descent number} of $u$ is $\cdes(u) := |\cDes(u)|$.
	A {\em cyclic word} $[\DAG u]\in \symm_A/\ZZ_a$ is an equivalence class of elements of $\symm_A$ under the cyclic equivalence relation 
	$(u_1,\ldots,u_a) \sim (u_{i+1}\ldots,u_a,u_1,\ldots,u_i)$ for all $i$.
	A {\em cyclic shuffle} of two cyclic 
	%permutations 
	words $[\DAG u]$ and $[\DAG v]$ with disjoint supports
	is the cyclic equivalence class $[\DAG w]$ represented by any shuffle $w$ of a representative of $[\DAG u]$ and a representative of $[\DAG v]$. The set of all cyclic shuffles of $[\DAG u]$ and $[\DAG v]$ is denoted $[\DAG u] \shuffle_{\cyc} [\DAG v]$,
	and is clearly a union of cyclic equivalence classes.
	
	%\todo{IG: $[\DAG a]$ and $[\DAG b]$ don't look good together; the arrows are at different heights. (For $[\DAG a]$ the arrow is inside the brackets but for  $[\DAG b]$ it is above the brackets.\\
	%RA: Done; $a$ and $b$ were replaced by $u$ and $v$.}
	
	The following cyclic analogue of Stanley's shuffling theorem~\cite[Ex.\ 7.93]{EC2} provides a combinatorial interpretation for the structure constants of $\cQSym^-$.
	
	\begin{theorem}\label{thm:main1}
		%\label{product-of-F-corollary}
		%Given a set partition $[n]=A\sqcup B$, 
		%$u$ in $\symm_A$ and $v$ in $\symm_B$, one has 
		%the following expansion
		%\[
		%F^\cyc_{|A|,[\cDes(u)]} \cdot F^\cyc_{|B|,[\cDes(v)]}
		%= \sum_{[\DAG{w}] \in [\DAG{u}] \shuffle_{\cyc} [\DAG{v}] }
		%  F^\cyc_{n,[\cDes(w)]}.
		%\]
		%\end{theorem}
		%\begin{theorem}
		%\label{product-of-F-corollary}
		Let $C = A \sqcup B$ be a disjoint union of finite sets of integers.
		For each $u \in \symm_A$ and $v \in \symm_B$, one has 
		the following expansion:
		\[
		F^\cyc_{|A|,[\cDes(u)]} \cdot F^\cyc_{|B|,[\cDes(v)]}
		= \sum_{[\DAG{w}] \,\, \in \,\, [\DAG{u}] \shuffle_{\cyc} [\DAG{v}] } F^\cyc_{|C|,[\cDes(w)]}.
		\]
	\end{theorem}
	
	%The proof is sketched in Section~\ref{sec:toric_posets} below.
	%See Theorem~\ref{product-of-F-corollary} below.
	
	%\bigskip
	
	%It is shown that 
	
	%\bigskip
	
	%\todo{IG: $\symm_n$, $\symm_A$, and $\shuffle_{\cyc}$ have not been defined (or introduced).\\
	%YR:Done}
	
	Recall that a skew shape is called a {\em ribbon} if it does not contain a $2\times  2$ square.
	%The following positivity phenomenon is proved in Section~\ref{sec:expansion}.
	
	\begin{theorem}\label{thm:main2}
		For every skew shape $\lambda/\mu$ which is not a connected ribbon, 
		all the coefficients in the expansion of the skew Schur function $s_{\lambda/\mu}$
		in terms of normalized fundamental cyclic quasi-symmetric functions are nonnegative integers. 
	\end{theorem}
	
	A more precise statement, which provides a combinatorial interpretation of the coefficients, is given in  %Theorem~\ref{conj:QS11} 
	\cref{t:Schur_in_hFcyc} below.
	The proof relies on the existence of a cyclic extension of the descent map on standard Young tableaux (SYT) of shape $\lambda/\mu$, which was proved in~\cite{ARR_CDes}. 
	Using 
	%Combining properties of 
	Postnikov's result regarding toric Schur functions, 
	%with results from~\cite{ARR_CDes}, 
	one deduces that the coefficients  in the expansion of a non-hook Schur function $s_\lambda$
	in terms of fundamental cyclic quasi-symmetric functions
	are equal to certain Gromov-Witten invariants.
	%, see Prop.~\ref{rem_Postnikov} below. 
	
	\smallskip
	
	%\todo{Define $\cDes$ and $\cdes$ of permutations}
	
	Applications to  
	%the distribution of cyclic descents on 
	the enumeration of SYT and cyclic shuffles of permutations with prescribed cyclic descent set or number follow from this theory. Using a ring homomorphism from $\cQSym$ to the ring of formal power series $\ZZ[[q]]_{\odot}$, with product defined by $q^i \odot q^j := q^{\max(i,j)}$,
	\cref{thm:main1} implies the following result.
	
	\begin{theorem}\label{thm:main3}
		Let $A$ and $B$ be two disjoint sets of integers with  $|A|=m$ and $|B|=n$.
		For each  $u\in \symm_A$ and $v\in \symm_B$ the following holds.
		%on disjoint sets of integers $U$ and $V$ respectively, with $|U|=m$ and $|V|=n$.
		\begin{itemize}
			\item[1.]
			If $\des(u) = i$ and $\des(v) = j$ then the number of shuffles of $u$ and $v$ with descent number $k$ is equal to
			\[
			\binom{m+j-i}{k-i} \binom{n+i-j}{k-j}.
			\]
			\item[2.] 
			If $\cdes(u) = i$ and $\cdes(v) = j$ then the number of cyclic shuffles of $[\DAG{u}]$ and $[\DAG{v}]$ with cyclic descent number $k$ is equal to
			\[
			%k \binom{m+j-i-1}{k-i} \binom{n+i-j-1}{k-j} + (m+n-k) \binom{m+j-i-1}{k-i-1} \binom{n+i-j-1}{k-j-1} \\
			\frac{k(m-i)(n-j) + (m+n-k)ij}{(m+j-i)(n+i-j)} \binom{m+j-i}{k-i} \binom{n+i-j}{k-j}. 
			%:= (m-i) \binom{m+j-i-1}{k-i-1} \binom{n+i-j}{k-j} + i \binom{m+j-i}%{k-i} \binom{n+i-j-1}{k-j}.
			\]
		\end{itemize}
	\end{theorem}
	
	%See Corollary~\ref{t:num_shuffles_given_cdes} below.
	%More enumerative applications are given in Section~\ref{sec:enumeration}.
	%%Theorem~\ref{conj22} and Proposition~\ref{prop:near-hooks} below.
	
	%\bigskip
	
	%\todo{YR: Add a paragraph regarding coproduct (Section~\ref{sec:co}).\\
	%RA: Done.}
	
	The group ring $\ZZ[\symm_n]$ has a distinguished subring, {\em Solomon's descent algebra} $\mathfrak{D}_n$, with basis elements
	\[
	D_I 
	:= \sum_{\substack{\pi \in \symm_n \\ \Des(\pi) = I}} \pi
	\qquad (I \subseteq [n-1]).
	\]
	Cellini~\cite{Cellini} and others 
	looked for an appropriate {\em cyclic} analogue. We provide a partial answer, using an operation dual to the product in $\mathfrak{D}_n$ --- the {\em internal coproduct} $\Delta_n$ on $\QSym_n$. 
	%see Section~\ref{sec:co} for its definition.
	%The analogue turns out to be a module, rather than an algebra.
	
	\begin{theorem}%\label{t:cQSym_right_coideal}
		$\cQSym_n$ and $\cQSym_n^-$ are right coideals of $\QSym_n$ with respect to the internal coproduct:
		\[
		\Delta_n(\cQSym_n) \subseteq \cQSym_n \otimes \QSym_n 
		%\qquad (\forall n \ge 0)
		\]
		and
		\[
		\Delta_n(\cQSym_n^-) \subseteq \cQSym_n^- \otimes \QSym_n. 
		%\qquad (\forall n \ge 0).
		\]
		The structure constants for $\cQSym_n^-$ are nonnegative integers.
		% Explicitly, in $\cQSym_n^-$:
		% % \[
		% % \Delta_n(F^{\cyc}_{n,B})
		% % = \sum_{\substack{A \in c2_{0,n}^{[n]} \\ J \subseteq [n-1]}}
		% % \tilde{a}_{B}^{AJ} F^{\cyc}_{n,A} \otimes F_{n,J}
		% % \qquad (\forall \,B \in c2_{0,n}^{[n]})
		% % \]
		% \[
		% \Delta_n(\hF^{\cyc}_{n,B})
		% = \sum_{\substack{A \in c2_{0,n}^{[n]} \\ J \subseteq [n-1]}}
		% \frac{d_A}{d_B} \tilde{a}_{B}^{AJ} \hF^{\cyc}_{n,A} \otimes F_{n,J}
		% \qquad (\forall \,B \in c2_{0,n}^{[n]}).
		% \]
		% where the number
		% \[
		% \tilde{a}_{B}^{AJ} := \#\{(\sigma_1, \sigma_2) \in \symm_n \times \symm_n \,:\, \cDes(\sigma_1) \in A,\, \Des(\sigma_2) = J,\, \sigma_2 \sigma_1 = \pi\}
		% \]
		% depends on $A, B \in c2_{0,n}^{[n]}$ and $J \subseteq [n-1]$ but not on $\pi$, as long as $\cDes(\pi) \in B$.
		% The structure constants $\frac{d_A}{d_B} \tilde{a}_{B}^{AJ}$ are nonnegative integers. 
		% In addition,
		% \[
		% \Delta_n(\hF^{\cyc}_{n,\varnothing}) 
		% = \Delta_n(h_n)
		% = \sum_{\lambda \vdash n} s_{\lambda} \otimes s_{\lambda}
		% \]
		% and
		% \[
		% \Delta_n(\hF^{\cyc}_{n,[n]}) 
		% = \Delta_n(e_n)
		% = \sum_{\lambda \vdash n} s_{\lambda} \otimes s_{\lambda'}
		% \]
		% belong to $\Sym_n \otimes \Sym_n \subseteq \cQSym_n \otimes \QSym_n$.
	\end{theorem}
	%\todo{RA: See ToDo after Theorem~\ref{t:cQSym_right_coideal}.
	%}
	%A more detailed description, including a combinatorial interpretation of the coefficients, 
	%see Theorem~\ref{t:cQSym_right_coideal}.
	%is given in the full version paper.
	
	\begin{corollary}%\label{t:left_module}
		For $n > 1$
		let $c2_{0,n}^{[n]}$ be the set of equivalence classes, under cyclic rotations, of subsets $\varnothing \subsetneq J \subsetneq [n]$.
		Defining
		\[
		cD_A
		:= \sum_{\substack{\pi \in \symm_n \\ \cDes(\pi) \in A}} \pi
		\qquad (A \in c2_{0,n}^{[n]}),
		\]
		the additive free abelian group
		\[
		\mathfrak{cD}_n := \spn_{\,\ZZ} \{cD_A \,:\, A \in c2_{0,n}^{[n]}\}
		\]
		is a left module for Solomon's descent algebra $\mathfrak{D}_n$.
	\end{corollary}
	%
	%In general, $\mathfrak{cD_n}$ is 
	%is not an algebra, namely, is not closed under the multiplication in $\ZZ[\symm_n]$.
	
	%\bigskip
	
	%The rest of the paper is organized as follows.
	
	%\bigskip
	
	%We conclude the paper with final remarks and open problems. 
	
	This is an extended abstract. 
	Proofs and more details are given in the full version of the paper~\cite{AGRR}.
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section{The fundamental cyclic quasi-symmetric functions}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\subsection{Fundamental cyclic quasi-symmetric functions}\label{sec:FCQSym}
	
	\begin{definition}\label{def:cyclic_word}
		For $n \ge 1$ and a subset $J \subseteq [n]$, denote by $P^{\cyc}_{n,J}$ the set of all pairs $(w,k)$ consisting of a word $w = (w_1, \ldots, w_n) \in \NN^n$ and an index $k \in [n]$ satisfying
		\begin{itemize}
			\item[(i)] 
			%The word $w$ is ``cyclically weakly increasing'' from index $k$, namely
			$w_k \le w_{k+1} \le \ldots \le w_n \le w_1 \le \ldots \le w_{k-1}$.
			%every pre-image $f^{-1}(m)$ forms a contiguous interval on the circle, and for every $a<b<c$ the intervals $f^{-1}(a), f^{-1}(b), f^{-1}(c)$ appear in clockwise order.
			\item[(ii)] 
			If $j \in J \setminus \{k-1\}$ then $w_j < w_{j+1}$, where indices are computed modulo $n$. 
			%(This condition is vacuous if $J = \varnothing$ or $J = \{k-1\}$.)
		\end{itemize}
	\end{definition}
	
	%\begin{remark}
	%The index $k$ is uniquely determined by the word $w$, unless all the letters of $w$ are equal; in which case, any index $k \in [n]$ will do. These ``constant words'' are relevant only for $J = \varnothing$, and the definition implies that each of them is counted $n$ times (and not just once) in $P^{\cyc}_{n,\varnothing}$.
	%\end{remark}
	
	\begin{example}
		Let $n=5$ and $J=\{1,3\}$. The pairs
		$(12345,1)$, $(23312,4)$ and $(23122,3)$ 
		%and $(23451,5)$ and $(21122,2)$
		%		$$
		%		23561,\   35612,\ 56123,\ 12211,\ 21122,\ 23311,\ 23312,\ 12233,\ 23122,\ 12241,\ 12311
		%		$$
		are in $P^{\cyc}_{5,\{1,3\}}$ (see \cref{fig:1}), 
		%- the set of cyclic $P$-partitions on a cycle of length $5$ with cyclic descent set $J=\{1,3\}$, 
		but the pairs $(12354,1)$, $(22312,4)$ and $(23112,3)$ are not. 
		%See Figure~\ref{fig:1} below.	
		
		\begin{figure}[htb]
			\begin{center}
				\begin{tikzpicture}[scale=0.35]
				\draw[red] (0,0) circle (4);
				\draw[fill] (69:4) circle (.1); \draw[blue](69:5) node%[above right]
				{$1$}; \draw[fill] (34.5:4) circle (.2);
				\draw[fill] (0:4) circle (.1); \draw[blue] (0:5) node%[right]
				{$2$};
				\draw[fill] (-69:4) circle (.1); \draw[blue] (-69:5) node%[below right]
				{$3$}; \draw[fill] (-106.5:4) circle (.2);
				\draw[fill] (-144:4) circle (.1); \draw[blue] (-144:5) node%[left]
				{$4$};
				\draw[fill] (144:4) circle (.1);\draw[blue] (144:5) node%[above left]
				{$5$};
				\draw[red] (180:4) node {$\wedge$};
				\end{tikzpicture}
				\ \ \ \ \ \
				\begin{tikzpicture}[scale=0.35]
				\draw[red] (0,0) circle (4);
				\draw[fill] (69:4) circle (.1); \draw[blue](69:5) node%[above right]
				{$2$}; \draw[fill] (34.5:4) circle (.2);
				\draw[fill] (0:4) circle (.1); \draw[blue] (0:5) node%[right]
				{$3$};
				\draw[fill] (-69:4) circle (.1); \draw[blue] (-69:5) node%[below right]
				{$3$}; \draw[fill] (-106.5:4) circle (.2);
				\draw[fill] (-144:4) circle (.1); \draw[blue] (-144:5) node%[left]
				{$1$};
				\draw[fill] (144:4) circle (.1);\draw[blue] (144:5) node%[above left]
				{$2$};
				\draw[red] (180:4) node {$\wedge$};
				\end{tikzpicture}
				\ \ \ \ \ \
				%	\end{center}
				%		\begin{center}
				\begin{tikzpicture}[scale=0.35]
				\draw[red] (0,0) circle (4);
				\draw[fill] (69:4) circle (.1); \draw[blue](69:5) node%[above right]
				{$2$}; \draw[fill] (34.5:4) circle (.2);
				\draw[fill] (0:4) circle (.1); \draw[blue] (0:5) node%[right]
				{$3$};
				\draw[fill] (-69:4) circle (.1); \draw[blue] (-69:5) node%[below right]
				{$1$}; \draw[fill] (-106.5:4) circle (.2);
				\draw[fill] (-144:4) circle (.1); \draw[blue] (-144:5) node%[left]
				{$2$};
				\draw[fill] (144:4) circle (.1);\draw[blue] (144:5) node%[above left]
				{$2$};
				\draw[red] (180:4) node {$\wedge$};
				\end{tikzpicture}
			\end{center}
			\caption{The pairs $(12345,1)$, $(23312,4)$ and $(23122,3)$ in $P^{\cyc}_{5,\{1,3\}}$.}
			\label{fig:1}
		\end{figure}		 
	\end{example}
	
	%\todo{Erase?
	\begin{defn}\label{def:Fcyc} 	
		Let $c2^{[n]}$ be the set of equivalence classes, under cyclic rotations, of subsets $\varnothing \subseteq J \subseteq [n]$.
		For any subset $J \subseteq [n]$ and orbit $A \in c2^{[n]}$
		define the {\em fundamental cyclic quasi-symmetric function} corresponding to $J$ or $A$ by 
		\[
		F^{\cyc}_{n,J}:=\sum_{(w,k) \in P^{\cyc}_{n,J}} x_{w_1} x_{w_2}\cdots x_{w_n} \quad \ \text{and}\ \quad F^{\cyc}_{n,A} := F^{\cyc}_{n,J}\ \ \  (\forall J\in A).
		\]
		%where $J$ is any element of the orbit $A$. 
		The corresponding {\em normalized fundamental cyclic quasi-symmetric function} is
		%$\hF^{\cyc}_{n,J}$ by %, as for the monomial functions,
		\[
		\hF^{\cyc}_{n,A} := \frac{1}{n} \sum_{J \in A} F^{\cyc}_{n,J}.
		%\qquad (\forall A \in c2^{[n]}).
		\]
	\end{defn}
	
	It is shown that these are all well-defined (i.e., independent of the choice of $J \in A$).
	
	\begin{proposition}\label{t:proper_basis}
		For each $n \ge 1$, the set $\left\{ \hF^{\cyc}_{n,A} \,:\, A \in c2^{[n]} \setminus \{[\emptyset]\} \right\}$ is a $\ZZ$-basis for $\cQSym_n$.
	\end{proposition}
	
	For many combinatorial applications
	%, in which the ``non-Escher'' property of cyclic descent sets (see Definition~\ref{def:cDes}) is manifest, 
	it is natural to consider a certain subring $\cQSym_n^-$ of $\cQSym_n$. 
	%\begin{defn}\label{def:cQSym-}
	%For any positive integer $n$, let $2_{0,n}^{[n]}$ be the set of all subsets of $[n]$ other than the empty set and $[n]$ itself.
	%Let $c2_{0,n}^{[n]}$ be the set of equivalence classes of elements of $2_{0,n}^{[n]}$ under cyclic shifts.
	Define %for each $n \ge 1$,
	\[
	\cQSym_n^- 
	:= \spn_{\ZZ} \left\{ \hF^{\cyc}_{n,A} \,:\, A \in c2^{[n]}\setminus\{[\emptyset],[[n]]\} \right\}
	%= \spn_{\ZZ} \left\{ \hF^{\cyc}_{n,J} \,:\, \varnothing \subsetneq J \subsetneq [n] \right\}.
	%\quad \cQSym_0^- := \ZZ,
	\qquad (n > 1),
	\]
	as well as $\cQSym_1^- 
	:= \spn_{\ZZ} \left\{ \hF^{\cyc}_{1,[[1]]} \right\}$,
	$\cQSym_0^- := \ZZ$, and
	$\cQSym^- := \bigoplus_{n \ge 0} \cQSym_n^-$.
	
	%In fact, $\cQSym$ is a ring and $\cQSym^-$ a subring; see Proposition~\ref{t:cQSym_is_a_ring}.
	%We call $\cQSym^-$ the {\em combinatorial subring} of $\cQSym$. 
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section{Toric posets and cyclic $P$-partitions}
	\label{sec:toric_posets}
	
	We recall {\em toric posets} from~\cite{DMR}, 
	%study cyclic analogues of posets, %that were called 
	and develop for them a theory of cyclic $P$-partitions.
	In particular, we provide a cyclic analogue of Stanley's fundamental decomposition lemma for $P$-partitions.
	%~\cite[Lemma 3.15.3]{EC1} 
	%will be given. % in Lemma~\ref{cyclic-fundamental-lemma}.
	Just as fundamental quasi-symmetric functions $F_{n,J}$ are $P$-partition enumerators for certain (labeled) total orders, 
	the fundamental cyclic quasi-symmetric functions
	$F^{\cyc}_{n,J}$ are cyclic $P$-partition enumerators for certain (labeled) total cyclic orders. 
	This is used to prove that $\cQSym^-$ is a ring and to study its structure constants.
	
	%\noindent
	%The following proposition explains why theory of DAGs is the same as that of posets.
	%\begin{proposition}
	%If $\DAG{D}_2$ extends $\DAG{D}_1$ then one has inclusions
	%\[
	%\begin{aligned}
	%\LLL(\DAG{D}_2) &\subseteq \LLL(\DAG{D}_1),\\
	%\AAA(\DAG{D}_2) & \subseteq \AAA(\DAG{D}_1),
	%\end{aligned}
	%\]
	%with equality in both cases
	%if $\DAG{D}_2$ is the transitive closure of $\DAG{D}_1$. 
	%\end{proposition}
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\subsection{Toric DAGs, toric posets, and toric $P$-partitions}\label{sec:toric}
	
	%The toric case is mildly trickier.
	%, because one must consider the following equivalence relation on DAGs.
	
	In this section, $\DAG{D}$ denotes a directed acyclic graph (DAG) with vertex set  $[n]:=\{1,2,\ldots,n\}$.
	Usual $P$-partitions use posets instead of DAGs, but the toric analogue will require DAGs.
	%DAGs will be our toric (or cyclic) analogues of posets.
	
	%a subset $\DAG{D}$
	%\begin{definition}
	A {\it $\DAG{D}$-partition} is a function $f\colon \{1,2,\ldots,n\} \rightarrow \{0,1,2,\ldots\}$ for which
	\begin{itemize}
		\item $f(i) \leq f(j)$ whenever $i \rightarrow j$ in $\DAG{D}$, and
		\item $f(i) < f(j)$ whenever $i \rightarrow j$ in $\DAG{D}$ but $i >_{\ZZ} j$.
	\end{itemize}
	Denote by $\AAA(\DAG{D})$ the set of all $\DAG{D}$-partitions $f$.
	%\end{definition}
	
	\begin{lemma}(Fundamental lemma of $\DAG{D}$-partitions \cite[Lemma 3.15.3]{EC1})
		\label{fundamental-lemma-of-P-partitions}
		For any DAG $\DAG{D}$, one has a decomposition of $\AAA(\DAG{D})$ as the following disjoint union:
		\[
		\AAA(\DAG{D}) = \bigsqcup_{ w \in \LLL(\DAG{D}) } \AAA(\DAG{w}),
		\]
		where $\LLL(\DAG{D})$ is the set of all linear (total) orders which extend $\DAG{D}$.
	\end{lemma}
	
	\begin{definition}
		%\begin{itemize}
		%\item 
		$(i)$ $i_0 \in [n]$
		is a {\it source} (respectively, {\it sink}) in $\DAG{D}$ if 
		$\DAG{D}$ contains no arrows of the form $j \rightarrow i_0$ (respectively, of the form $i_0 \rightarrow j$). 
		
		%\item 
		\noindent
		$(ii)$ $\DAG{D'}$
		is obtained from $\DAG{D}$ by a %n {\it elementary equivalence} or 
		{\it flip at $i_0$} if $i_0$ is either a source or a sink of $\DAG{D}$ and one obtains $\DAG{D}'$ by reversing all the arrows in $\DAG{D}$ incident with $i_0$. 
		
		%\item 
		\noindent
		$(iii)$ Define the equivalence relation $\equiv$ on DAGs to be
		the reflexive-transitive closure of the %elementary equivalences
		flips, that is, $\DAG{D} \equiv \DAG{D'}$ if and only if there exists a (possibly empty) sequence of flips one can apply starting with $\DAG{D}$ to obtain $\DAG{D'}$.
		%\end{definition}
		%\begin{definition}
		%\item 
		
		\noindent
		$(iv)$ A {\it toric DAG} is the $\equiv$-equivalence class $[\DAG{D}]$ of a DAG $\DAG{D}$.
		%\end{itemize}
	\end{definition}
	
	\begin{example}
		Here is an example of a toric DAG $[\DAG{D}_1]$:
		%\begin{figure}
		\[
		\xymatrix@R=12pt{
			4                & \\
			& 3 \ar[ul] \\
			& 2 \ar[u] \\
			1\ar[ur] \ar[uuu]
		} 
		\xymatrix@R=12pt{
			1                & \\
			& 4 \ar[ul] \\
			& 3 \ar[u] \\
			2\ar[ur] \ar[uuu]
		}
		\xymatrix@R=12pt{
			2                & \\
			& 1 \ar[ul] \\
			& 4 \ar[u] \\
			3\ar[ur] \ar[uuu]
		} 
		\xymatrix@R=12pt{
			3                & \\
			& 2 \ar[ul] \\
			& 1 \ar[u] \\
			4\ar[ur] \ar[uuu]
		}
		\]
		%\end{figure}
		
		Here is another toric DAG $[\DAG{D}_2]$:
		{\footnotesize
		$$
		\xymatrix@R=12pt{
			& 1              & \\
			2\ar[ur]&                &4\ar[ul] \\
			& 3\ar[ur]\ar[ul]&
		} \quad
		\xymatrix@R=12pt{
			2              & 4 \\
			1\ar[u]\ar[ur] & 3\ar[u]\ar[ul]
		}\quad
		\xymatrix@R=12pt{
			& 2              & \\
			1\ar[ur]&                &3\ar[ul] \\
			& 4\ar[ur]\ar[ul]&
		}\quad
		\xymatrix@R=12pt{
			& 3              & \\
			2\ar[ur]&                &4\ar[ul] \\
			& 1\ar[ur]\ar[ul]&
		}\quad
		\xymatrix@R=12pt{
			1              & 3 \\
			2\ar[u]\ar[ur] & 4\ar[u]\ar[ul]
		}\quad
		\xymatrix@R=12pt{
			& 4              & \\
			1\ar[ur]&                &3\ar[ul] \\
			& 2\ar[ur]\ar[ul]&
		}
		$$}
	\end{example}
	
	
	\begin{definition}
		Say that $[\DAG{D}_2]$ {\it torically extends} $[\DAG{D}_1]$ if there exist $\DAG{D}_i' \in [\DAG{D}_i]$ for $i=1,2$ with 
		$\DAG{D'_1} \subseteq \DAG{D'_2}$.
	\end{definition}
	
	A certain toric extension, called the toric transitive closure, will be particularly important.
	\begin{definition}
		$(i)$ Say that $i \rightarrow j$ is implied from {\it toric transitivity} in a DAG $\DAG{D}$ if there exist in $\DAG{D}$ 
		both a chain $i_1 \rightarrow  i_2 \rightarrow \cdots \rightarrow i_k$ and a direct arrow $i_1 \rightarrow i_k$ 
		such that $i=i_a, j=i_b$ for some $1 \leq a < b \leq k$.
		
		\noindent
		$(ii)$ The {\em toric transitive closure} of $\DAG{D}$ is the DAG $\DAG{P}$ obtained by adding in all arrows $i \rightarrow j$ implied from toric transitivity in $\DAG{D}$.
		%;
		%see Figure~\ref{fig:toric-transitive-closure}, where the existence of all the solid arcs in $\DAG{D}$ implies the existence of all the dotted arcs in its toric transitive closure.
		%Note that the toric transitive closure of $\DAG{D}$ is a subset of its usual transitive closure. 
		
		\noindent
		$(iii)$ A DAG $\DAG{D}$ is {\em toric transitively closed} if it equals its toric transitive closure.
	\end{definition}
	
	%\begin{figure}
	%\[
	%\xymatrix@R=20pt{                                                                                                     
	%i_k & & & &\\                                                                                                       
	%                    & &i_{k-1} \ar[ull]& &\\                                              %                          
	%                   & & &i_{k-2} \ar[ul]\ar@{-->}[uulll] &\\                               %                            
	%                    & & & &\vdots \ar[ul]\ar@{-->}[uuullll] \\                                                                          
	%                    & & &i_3 \ar[ur]\ar@{-->}[uuuulll] &\\                                                             
	%                    & &i_2 \ar[ur] \ar@{-->}[uuuuull] \ar@{-->}[uuur]\ar@{-->}[uuuu]&  & \\                                                           
	%i_1 \ar[uuuuuu] \ar[urr] %\ar@{-->}[uurrr]\ar@{-->}[uuuurrr]\ar@{-->}[uuuuurr]\ar@{-->}[uuurrrr]&  &  &  &    }    
	%\]
	%\caption{Toric transitive closure}
	%\label{fig:toric-transitive-closure}
	%\end{figure}
	
	\begin{proposition}
		\label{toric-closure-independent-of-representative}
		If $\DAG{D_1} \equiv \DAG{D_2}$, then $\DAG{D}_1$ is toric transitively closed if and only if so is $\DAG{D}_2$.
	\end{proposition}
	
	
	\begin{definition}
		A toric DAG $[\DAG{D}]$ is a {\em toric poset}
		%\footnote{This isn't quite the way that it was defined in \cite{DMR}, but essentially equivalent, via \cite[Theorem 1.4]{DMR}.} 
		if $\DAG{D}$ is toric transitively closed for one of its 
		$\equiv$-class representatives  $\DAG{D}$, or equivalently, by \cref{toric-closure-independent-of-representative}, for {\it all} such %representatives 
		$\DAG{D}$. 
	\end{definition}
	
	\begin{definition}
		A {\em total cyclic order} is 
		%a toric poset $[\DAG{w}]$ for some $w=(w_1,\ldots,w_n)$ in $\symm_n$, that is,
		%\[
		%\begin{aligned}[]
		%[\DAG{w}] =
		%&\{ (w_1,w_2,\ldots,w_{n-1},w_n)^\rightharpoonup, \\
		%&\,\,\, (w_2,w_3,\ldots,w_{n},w_1)^\rightharpoonup, \\
		%&\,\,\, \DAG{(w_3,w_4,\ldots,w_n, \,\, w_1, \,\, w_2)}, \\
		%&\,\,\, \qquad \qquad \vdots \\
		%&\,\,\, (w_{n},w_1,\ldots,w_{n-2},w_{n-1})^\rightharpoonup\}.
		%\end{aligned}
		%\]
		%In other words, a total cyclic order is 
		a toric poset with
		at least one (equivalently,  all) of its $\equiv$-class
		representatives being a total (linear) order.
		
		Denote by $\LLL^\tor([\DAG{D}])$ the set of all total cyclic orders $[\DAG{w}]$ which torically extend $[\DAG{D}]$.
	\end{definition}
	
	\begin{remark}\label{rem:cyclic_perm}%\ \\
		% $[\DAG{w}]$, where $w=(w_1,\ldots,w_n)\in \symm_n$,
		%\begin{itemize}
		%\item[1.]
		%A total cyclic order $[\DAG{w}]$ may be viewed as a coset $w\ZZ_n \in \symm_n/\ZZ_n$, where $\ZZ_n$
		%is the subgroup of $\symm_n$ generated by the $n$-cycle $(2,3,\dots,n,1)$.
		%\item[2.]
		%Total cyclic orders may be identified with $n$-cycles in $\symm_n$.
		%\item[3.]
		Total cyclic orders %is sometimes called a
		may be %identified with 
		geometrically visualized as
		$n$ dots in a directed cycle labeled 
		by $1,\dots,n$ with no repeats. These configurations are called {\em cyclic permutations},
		and will be used in the study of cyclic shuffles, see \cref{fig:cyc_shuffle}.
		
		%$[\DAG{w}]$ 
		%since it is an equivalence class of permutations identified up to 
		%rotation: $w_1,\dots,w_n\sim w_{i+1}\dots,w_n,w_1,\dots,w_i$.
		%\end{itemize}
	\end{remark}
	
	\begin{definition}
		A {\em toric $[\DAG{D}]$-partition} is a function $f\colon \{1,2,\ldots,n\} \rightarrow \{0,1,2,\ldots\}$ which
		is a $\DAG{D}'$-partition for at least one DAG $\DAG{D}'$ in $[\DAG{D}]$.
		Let  $\AAA^\tor([\DAG{D}])$ denote the set of all toric $[\DAG{D}]$-partitions \end{definition}
	
	
	\begin{lemma}(Fundamental lemma of toric $\DAG{D}$-partitions)
		\label{cyclic-fundamental-lemma}
		For any DAG $\DAG{D}$, one has a decomposition of $\AAA^\tor([\DAG{D}])$ as the following disjoint union:
		\[
		\AAA^\tor([\DAG{D}]) 
		= \bigsqcup_{ [\DAG{w}] \in \LLL^\tor([\DAG{D}]) } \AAA^\tor([\DAG{w}]).
		\]
	\end{lemma}
	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\subsection{Cyclic $P$-partition enumerators}
	%\label{sec:CycP}
	
	\begin{definition}
		Given a toric poset $[\DAG{D}]$ on $\{1,2,\ldots,n\}$, 
		define its cyclic $P$-partition enumerator
		\[
		F^\cyc_{[\DAG{D}]}:=
		\sum_{f \in \AAA^\tor([\DAG{D}])} 
		x_{f(1)} x_{f(2)} \cdots x_{f(n)}.
		\]
	\end{definition}
	
	
	A special case %of these enumerators 
	yields the fundamental cyclic quasi-symmetric functions
	from \cref{def:Fcyc}. 
	%defined in Subsection~\ref{sec:FCQSym}.
	%Recall the notations $P^{\cyc}_{n,J}$ and $F^\cyc_{n,J}$.
	%set $P^{\cyc}_{n,J}$ from Definition~\ref {def:cyclic_word}.
	
	\begin{proposition}
		If $w \in \symm_n$ has $\cDes(w)=J$, then 
		%\[
		%\AAA^\tor([\DAG{w}]) =\{f\in \NN^n:\ \exists i\ (f,i)\in P^{\cyc}_{n,J} \}.
		%\]
		%Consequently, 
		%\[
		$F^\cyc_{[\DAG{w}]} = F^\cyc_{n,J}$.
		%\]
		%In particular,  this shows that $F^\cyc_{[\DAG{w}]}$ lies in $\cQSym^-$.
	\end{proposition}
	
	%\todo{RA: Note that the definition of $P^{\cyc}_{n,J}$ has changed; it now consists of pairs $(w,k)$. See Definition~\ref{def:cyclic_word}.\\
	%YR: handled.}
	
	
	
	
	An immediate consequence of 
	\cref{cyclic-fundamental-lemma} is then the following.
	
	\begin{proposition}
		\label{cyclic-F-expansion}
		For any toric poset $[\DAG{D}]$,
		one has the following expansion
		\[
		F^\cyc_{[\DAG{D}]}
		=\sum_{[\DAG{w}] \in \LLL^\tor([\DAG{D}])}
		F^\cyc_{n,\cDes(w)}.
		\]
		%In particular, this shows that $F^\cyc_{[\DAG{D}]}$ lies in $\cQSym^-$.
	\end{proposition}
	
	We now use this fact to expand products of
	of basis elements $\{F^\cyc_{n,J}\}$ back in the same
	basis.  The key notion is that of a cyclic shuffle
	of two total cyclic orders.
	
	\medskip
	
	First recall %the well known non-cyclic analogue.
	the notion of a shuffle of permutations.
	%Denote the symmetric group on a a finite set of integers $A$ by $\symm_A$.
	For a finite set $A$ of size $a$, let $\symm_A$ be 
	the set of all bijections $w \colon [a] \to A$,
	viewed as words $w = (w_1, \ldots, w_a)$.
	%If $A = [a]$ then $\symm_A$ is of course the symmetric group $\symm_a$. 
	%%By a slight abuse of language, we shall call the elements of $\symm_A$ ``permutations''. 
	Elements of $\symm_A$ will be called {\em bijective words}, a formal extension of permutations.
	%If $A$ is a set of integers, or any totally ordered set, define a {\em descent set} for each $w \in \symm_A$ by
	%\[
	%\Des(w) := \{i \in [a-1] \,:\, w(i) > w(i+1)\}.
	%\]
	%\todo{IG: I don't think that this definition of $\symm_A$ is correct. The  symmetric group on $A$ is the set of bijections from $A$ to itself. What we want is the set of bijections from $[a]$ to $A$, viewed as words or sequences of length $a$, where $a=|A|$.
	%RA: Done.}
	Given two bijective words $u = (u_1,\ldots,u_a)\in \symm_A$ and $v = (v_1,\ldots,v_b)\in \symm_B$, 
	where $A$ and $B$ are disjoint finite sets of integers, 
	a bijective word $w \in \symm_{A \sqcup B}$ is a {\em shuffle} of $u$ and $v$ 
	if $u$ and $v$ are subwords of $w$.
	Denote the set of all shuffles of $u$ and $v$ by 
	%$\Sh(u,v)$.
	$u \shuffle v$.
	%The following theorem is a straightforward generalization of~\cite[Ex.\ 7.93]{EC2}.
	
	%\begin{theorem}%[{see~\cite[Ex.\ 7.93]{EC2}}]
	%\label{non-cyclic-product-of-F}
	%Let $C = A \sqcup B$ be a disjoint union of finite sets of integers.
	%For each $u \in \symm_A$ and $v \in \symm_B$, one has 
	%the following expansion:
	%\[
	%F_{|A|,\Des(u)} \cdot F_{|B|,\Des(v)}
	%= \sum_{w \in u\shuffle v } F_{|C|,\Des(w)}.
	%\]
	%\end{theorem}
	
	
	\begin{definition}
		Let $C = A \sqcup B$ be a 
		%set partition.
		disjoint union of finite sets.
		%, and denote $a:=|A|$ and $b:=|B|$.
		Fix two total cyclic orders $[\DAG{u}]$ and $[\DAG{v}]$,
		with representatives $u = (u_1,\ldots,u_a)\in \symm_A$
		% - the symmetric group on $A$ 
		and $v = (v_1,\ldots,v_b)\in \symm_B$. 
		%- the symmetric group on $B$. 
		%, and let $n:=a+b$.
		A total cyclic order $[\DAG{w}]$, with $w \in \symm_C$, is a {\em cyclic shuffle of $[\DAG{u}]$ and $[\DAG{v}]$}
		%, written $[\DAG{w}] \in [\DAG{u}] \shuffle_{\cyc} [\DAG{v}]$,
		if there exists a representative $w' \in \symm_C$ of $[\DAG{w}]$
		%such that the restrictions of $w$ to the sets of values $\{1,2,\ldots,a\}$ and $\{a+1,a+2,\ldots,a+b\,(=n)\}$, respectively, are 
		which is 
		(equivalently, every representative of $[\DAG{w}]$ is) a shuffle of cyclic shifts of $u$ and $v$, namely,
		\[
		w' \in u' \shuffle v'
		\]
		for some cyclic shift $u'$ of $u$ and cyclic shift $v'$ of $v$.
		% \[
		% \begin{array}{ccccc}
		% u&=&(u_1,u_2,\ldots,u_a)&=&(w'_{i_1},w'_{i_2},\ldots,{w'_{i_a}}),\\
		% &&&&\\
		% v+(a,\ldots,a)&=&(v_1+a,v_2+a,\ldots,v_b+j)
		% &=&(w''_{j_1},w''_{j_2},\ldots,w''_{j_b})
		% \end{array}
		% \]
		% for some $1 \leq i_1 < \cdots < i_a \leq n$
		% and $1 \leq j_1 < \cdots < j_b \leq n$.
		
		Denote the set of all cyclic shuffles of $[\DAG u]$ and $[\DAG v]$ by 
		%$\Sh(u,v)$.
		$[\DAG u]\shuffle_\cyc [\DAG v]$.
	\end{definition}
	
	%\begin{remark}
	%Recalling Remark~\ref{rem:cyclic_perm},
	%a cyclic permutation $[\DAG{w}]\in \symm_n/\ZZ_n$ is a 
	%{\em cyclic shuffle of $[\DAG{u}]$ and $[\DAG{v}]$} if the cyclic subsequence
	%of $[\DAG{w}]$ consisting the letters from $U$ is the cyclic permutation 
	%$[\DAG{u}]$ and the cyclic subsequence
	%of $[\DAG{w}]$ consisting the letters from $V$ is the cyclic permutation 
	%$[\DAG{v}]$.
	%\end{remark}
	
	%\begin{definition}
	%Fix two total cyclic orders $[\DAG{u}]$ and $[\DAG{v}]$,
	%with representatives $u = (u_1,\ldots,u_a) \in \symm_a$ and $v = %(v_1,\ldots,v_b) \in \symm_b$, and let $n:=a+b$.
	%A total cyclic order $[\DAG{w}]$, with $w \in \symm_n$, is a {\em cyclic %shuffle of $[\DAG{u}]$ and $[\DAG{v}]$}, written $[\DAG{w}] \in [\DAG{u}] %\shuffle_{\cyc} [\DAG{v}]$,
	%if there exists a representative $w' \in \symm_n$ of $[\DAG{w}]$
	%%such that the restrictions of $w$ to the sets of values $\{1,2,\ldots,a\}$ %and $\{a+1,a+2,\ldots,a+b\,(=n)\}$, respectively, are 
	%which is 
	%(equivalently, every representative of $[\DAG{w}]$ is) a shuffle of cyclic %shifts of $u$ and $v+(a,\ldots,a)$:
	%\[
	%w' = u' \shuffle (v'+(a,\ldots,a))
	%\]
	%where $u'$ is a cyclic shift of $u$ and $v$ is a cyclic shift of $v$.
	%% \[
	%% \begin{array}{ccccc}
	%% u&=&(u_1,u_2,\ldots,u_a)&=&(w'_{i_1},w'_{i_2},\ldots,{w'_{i_a}}),\\
	%% &&&&\\
	%% v+(a,\ldots,a)&=&(v_1+a,v_2+a,\ldots,v_b+j)
	%% &=&(w''_{j_1},w''_{j_2},\ldots,w''_{j_b})
	%% \end{array}
	%% \]
	%% for some $1 \leq i_1 < \cdots < i_a \leq n$
	%% and $1 \leq j_1 < \cdots < j_b \leq n$.
	%\end{definition}
	
	\begin{example}
		\label{cyclic-shuffle-example}
		Let $A=\{1,3,5,7,8\}$ and $B=\{2,4,6,9\}$,
		%(so that $n=9$), 
		and fix
		$u = (3,7,8,5,1) \in \symm_A$ and $v = (6,9,4,2) \in \symm_B$.
		%Then $v+(a,a,a,a)=(9,8,7,6)$, and an 
		An example of
		$[\DAG{w}] \in [\DAG{u}] \shuffle_{\cyc} [\DAG{v}]$
		is 
		$[(8,4,5,1,2,3,6,7,9)]$, since $w' = (1,2,3,6,7,9,8,4,5)$ is a shuffle of $(1,3,7,8,5)\in [\DAG{u}]$ and $(2,6,9,4)\in [\DAG{v}]$. See \cref{fig:cyc_shuffle}.
		%the one having either these two representatives $w',w''$, which contain $u,v+(a,a,a,a)$ as subsequences:
		%\[
		%\begin{aligned}
		%w'&=(\mathbf{1},9, \mathbf{2},\mathbf{3},8,7,\mathbf{4},\mathbf{5},6)\\
		%w''&=(\mathbf{9},2,3,\mathbf{8},\mathbf{7},4,5,\mathbf{6},1)
		%\end{aligned}
		%\]
	\end{example}
	
	\begin{figure}
		\begin{center}
			
			\begin{tikzpicture}[scale=0.25]
			\draw[red] (0,0) circle (5);
			\draw[fill] (90:5) circle (.15); \draw[blue](90:6) node%[above right]
			{$8$}; 
			\draw[fill] (50:5) circle (.15); \draw[blue](50:6) node%[above right]
			{$4$}; 
			\draw[fill] (10:5) circle (.15); \draw[blue](10:6) node%[above right]
			{$5$}; 
			\draw[fill] (330:5) circle (.15); \draw[blue](330:6) node%[above right]
			{$1$}; 
			\draw[fill] (290:5) circle (.15);\draw[blue] (290:6) node%[above left]
			{$2$};
			\draw[fill] (250:5) circle (.15);\draw[blue] (250:6) node%[above left]
			{$3$};
			\draw[fill] (210:5) circle (.15);\draw[blue] (210:6) node
			{$6$};
			\draw[fill] (170:5) circle (.15);\draw[blue] (170:6) node
			{$7$};
			\draw[fill] (130:5) circle (.15);\draw[blue] (130:6) node
			{$9$};
			\draw[red] (180:5) node {$\wedge$};
			\draw[black] (0:9.5) node {$\in$};
			\end{tikzpicture}
			\ \ \ 
			\begin{tikzpicture}[scale=0.25]
			\draw[red] (0,0) circle (5);
			\draw[fill] (90:5) circle (.15); \draw[blue](90:6) node
			{$3$}; 
			\draw[fill] (18:5) circle (.15); \draw[blue](18:6) node
			{$7$}; 
			\draw[fill] (306:5) circle (.15);\draw[blue] (306:6) node
			{$8$};
			\draw[fill] (234:5) circle (.15);\draw[blue] (234:6) node
			{$5$};
			\draw[fill] (162:5) circle (.15);\draw[blue] (162:6) node
			{$1$};
			
			\draw[red] (180:5) node {$\wedge$};
			\end{tikzpicture}
			\ \ \
			\begin{tikzpicture}[scale=0.25]
			\draw[red] (0,0) circle (5);
			\draw[fill] (45:5) circle (.15); \draw[blue](54:6) node {$6$}; 
			
			\draw[fill] (315:5) circle (.15); \draw[blue](306:6) node {$9$}; 
			
			\draw[fill] (225:5) circle (.15);\draw[blue] (234:6) node {$4$};
			
			\draw[fill] (135:5) circle (.15);\draw[blue] (126:6) node {$2$};
			
			\draw[red] (180:5) node {$\wedge$};
			
			\draw[black] (180:10) node {$\shuffle_{\cyc}$}; 
			%\draw[black] (0:9) node {$=$};
			\end{tikzpicture}
		\end{center}
		\caption{$[(8,4,5,1,2,3,6,7,9)]\in [(3,7,8,5,1)]\shuffle_\cyc [(6,9,4,2)]$.}
		\label{fig:cyc_shuffle}
	\end{figure}
	
	\begin{observation}
		\label{number-of-cyclic-shuffles}
		Let $A$ and $B$ be disjoint sets of integers, of cardinalities $a$ and $b$ respectively.
		For each $u=(u_1,u_2,\ldots,u_a) \in \symm_A$ and $v=(v_1,v_2,\ldots,v_b) \in \symm_B$ there are
		$\frac{(a+b-1)!}{(a-1)!(b-1)!}$ cyclic shuffles %$[\DAG{w}]$ 
		in $[\DAG{u}] \shuffle_{\cyc} [\DAG{v}]$.
	\end{observation}
	
	We apply this setting to prove \cref{thm:main1} and deduce the following consequences.
	
	%the following cyclic analogue of Theorem~\ref{non-cyclic-product-of-F}.
	
	%\begin{theorem}
	%\label{product-of-F-corollary}
	%%Given a set partition $[n]=A\sqcup B$, 
	%Let $C = A \sqcup B$ be a disjoint union of finite sets of integers.
	%For each $u \in \symm_A$ and $v \in \symm_B$, one has 
	%the following expansion:
	%\[
	%F^\cyc_{|A|,\cDes(u)} \cdot F^\cyc_{|B|,\cDes(v)}
	%= \sum_{[\DAG{w}] \in [\DAG{u}] \shuffle_{\cyc} [\DAG{v}] } F^\cyc_{|C|,\cDes(w)}.
	%\]
	%\end{theorem}
	
	%\todo{For enumerative applications, we need a generalized version of this Theorem,
	%where $u$ and $v$ are any two permutations with complementary support}
	
	
	
	%\todo{RA: Note that the $\ZZ$-basis elements of $\cQSym$ are actually $d_A^{-1} F^\cyc_{n,A}$ for the orbits $A \in \cnes$; see Lemma~\ref{t:Fcyc_properties}.\\
	%}
	%\todo{
	%YR: The above method is applicable for computation of the structure constants of the {\em normalized} fundamental cyclic QSF as well, but result is less transparent.\\
	%Question: Is there an elegant {\em direct} combinatorial interpretation for these structure constants of the {\em normalized} fundamental cyclic QSF ?}
	
	%Let us now return to prove that $\cQSym$ is a ring. It will helpful to coefficients be in either ring $R=\ZZ$ or $\QQ$.
	%%, so let 
	%Let $R[\xx]_{\bded}$ denote the subring of $R[[x_1,x_2,\ldots]]$ consisting of the formal power series $f(x_1,x_2,\ldots)$ having bounded degree and  let $\cQSym_R$ denote the
	%$R$-submodule of $R[\xx]_{\bded}$ consisting of those $f$ cyclically quasi-symmetric as in Definition~\ref{def:cQSym}; that is,
	%for any $t \ge 1$, any two increasing sequences 
	%$i_1 < \dots < i_t$
	%and $i'_1 < \dots < i'_t$ of positive integers, any sequence
	%$m = (m_1, \ldots, m_t)$ of positive integers, and any cyclic shift
	%$m' = (m'_1, \ldots, m'_t)$ of $m$, the coefficients of
	%$x_{i_1}^{m_1} \cdots x_{i_t}^{m_t}$ and 
	%$x_{i'_1}^{m'_1} \cdots x_{i'_t}^{m'_t}$ in $f$ are equal.
	
	\begin{proposition}
		\label{t:cQSym_is_a_ring}
		%For either choice of coefficient ring $R=\ZZ, \QQ$, the cyclic quasi-symmetric functions 
		$\cQSym$ and $\cQSym^-$ are graded rings.
		%, that is, they are closed under multiplication.
	\end{proposition}
	%\todo{RA: What about the structure constants of $\cQSym$?}
	
	\begin{proposition}\label{t:nonnegative_structure_constants}
		The structure constants of $\cQSym$ and $\cQSym^-$, with respect to the normalized fundamental basis, are nonnegative integers.
	\end{proposition}
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section{Expansion of Schur functions in terms of fundamental cyclic quasi-symmetric functions}\label{sec:expansion}
	
	%Recall that the ring of symmetric functions, $\Sym$, is  a graded subring of
	% $\cQSym$. %\end{obs}
	%%In this section we prove the following theorem.
	
	%\begin{theorem}\label{conj:QS11}
	%For every skew shape $\lambda/\mu$ which is not a connected ribbon, 
	%all the coefficients in the expansion of the skew Schur function $s_{\lambda/\mu}$
	%in terms of normalized fundamental cyclic quasi-symmetric functions are nonnegative integers. 
	%\end{theorem}
	
	\cref{thm:main2} follows from %Theorem~\ref{conj:QS1} 
	\cref{t:Schur_in_hFcyc} 
	below.
	The {\em cyclic descent map} on SYT of a given shape plays a key role in the proof; 
	let us recall the relevant definition and main result from \cite{ARR_CDes}.
	
	\begin{definition}[{\cite[Definition 2.1]{ARR_CDes}}]\label{def:cDes}
		Let $\TTT$ be a finite set, equipped with 
		%an arbitrary map (called 
		a {\em descent map}
		$\Des\colon \TTT \longrightarrow 2^{[n-1]}$,
		where $n > 1$.
		A {\em cyclic extension} of $\Des$ is
		a pair $(\cDes,p)$, where 
		$\cDes\colon \TTT \longrightarrow 2^{[n]}$ is a map 
		and $p\colon \TTT \longrightarrow \TTT$ is a bijection,
		satisfying the following axioms:  for all $T$ in $\TTT$:
		\[
		\begin{array}{rl}
		\text{(extension)}   & \cDes(T) \cap [n-1] = \Des(T),\\
		\text{(equivariance)}& \cDes(p(T))  = 1 + \cDes(T),\\
		\text{(non-Escher)}  & \varnothing \subsetneq \cDes(T) \subsetneq [n].\\
		\end{array}
		\]
	\end{definition}
	
	\begin{example}
		Let $\TTT$ be $\symm_n$, the symmetric group on $n$ letters equipped with the classical descent map.
		The pair $(\cDes,p)$, 
		with $\cDes$ defined as in \eqref{e.cellini} and $p$ the cyclic shift, 
		satisfies the axioms of \cref{def:cDes}.
	\end{example}
	
	%\todo{RA: Define $\Des$ for SYT.}    
	
	%\medskip
	The notion 
	of a descent set for a {\em standard Young tableau} $T$ of skew shape $\lambda/\mu$
	is well established (see, e.g., \cite[p.~361]{EC2}) .
	%(and size $n$):
	%\[
	%\Des(T) := \{1 \le i \le n-1 \,:\,
	%            i+1 \text{ appears in a lower row of }T\text{ than }i\}
	%\quad \subseteq [n-1].
	%\]
	%For example, the following standard Young tableau $T$ of shape $\lambda/\mu=(4,3,2)/(1,1)$ 
	%has $\Des(T)=\{2,3,5\}$:
	%\[
	%\large\young(:127,:35,46)
	%\]
	For the special case of {\em rectangular} shapes,
	Rhoades~\cite{Rhoades} 
	%introduced a notion of {\em cyclic descent set} $\cDes(T)$, 
	%having the same properties\eqref{restriction-property}, \eqref{equivariance-property} and \eqref{non-Escher-property}
	constructed a cyclic extension %$(\cDes,p)$ of $\Des$ 
	satisfying the axioms of \cref{def:cDes}.
	%where $p$ acts on SYT via Sch\"utzenberger's {\em jeu-de-taquin promotion} operator. 
	For almost all skew shapes there is a general existence result, as follows.
	
	\begin{theorem}[{\cite[Theorem 1.1]{ARR_CDes}}]\label{t:cyclic_extension}
		Let $\lambda/\mu$ be a skew shape with $n$ cells.
		The descent map $\Des$ on $\SYT(\lambda/\mu)$ has a cyclic extension $(\cDes,p)$ if and only if $\lambda/\mu$ is not a connected ribbon.
		Furthermore, for all $J \subseteq [n]$, all such cyclic extensions share the same cardinalities $\#{\cDes^{-1}(J)}$.
	\end{theorem}
	
	A constructive combinatorial proof of \cref{t:cyclic_extension} was recently given in~\cite{Huang}.    
	
	%\medskip
	
	We shall now provide %, in Theorem~\ref{t:Schur_in_hFcyc},
	a cyclic analogue of the classical result 
	\cite[Theorem 7.19.7]{EC2} (first proved in~\cite[Theorem 7]{Gessel}).
	
	
	
	%Part 1 of the conjecture was checked on all non-hook straight shapes of size $n<7$.
	
	
	%The following corollary shows that the coefficients in the expression of a skew Schur function in terms of the basis of {\em normalized} fundamental cyclic quasi-symmetric functions are equal to the corresponding fiber sizes of (any) cyclic descent map.
	
	\begin{theorem}\label{t:Schur_in_hFcyc}
		%Under the assumptions of Theorem~\ref{conj:QS1},
		For every skew shape $\lambda/\mu$ of size $n$, which is not a connected ribbon, and for any %cyclic descent map $\cDes$ 
		cyclic extension $(\cDes,p)$ of $\Des$ 
		on $\SYT(\lambda/\mu)$,
		\[
		s_{\lambda/\mu}
		= \sum_{A \in c2_{0,n}^{[n]}} m^{\cyc}(A) \,\hF^{\cyc}_{n,A}
		\]
		%where $s_{\lambda/\mu}$ is the Schur function indexed by $\lambda/\mu$.
		where
		\[
		m^{\cyc}(A) := m^{\cyc}(J)
		= \# \{T \in \SYT(\lambda/\mu) \,:\, \cDes(T) =J\}
		\qquad \left( \forall J \in A \in c2_{0,n}^{[n]} \right).
		\]
		%This number is independent of the choice of $J$ in $A$. In fact,
		%\[
		%m^{\cyc}(A) 
		%= \frac{1}{\# A} \sum_{J \in A} m^{\cyc}(J)
		%= \frac{1}{\# A} \cdot \# \{T \in \SYT(\lambda/\mu) \,:\, \cDes(T) \in A\}.
		%\]
	\end{theorem}
	
	%\todo{Erase?
	
	%{\bf Proof Idea.}
	%First, by Remark~\ref{rem:QS1}, we may assume that $n\in J$.
	
	%Second, every cyclic weakly increasing word which respects $J$, 
	%which is supported by a subset of letters in $[n]$, 
	%may be obtained by a cyclic $\ZZ_n$-shift of the values of the letters of a unique weakly increasing %word which have strict increase at positions in $J$. Thus it is the content of a SYT $T\in %\SYT(\lambda/\mu)$ with $\Des(T)=J\setminus \{n\}$
	%(with respect to some rotated order $i+1<i+2<\cdots<n<1<\cdots< i$).
	%Using the combinatorial definition of Schur functions and symmetry,
	%one can deduce that summing over all words which respect same rotated order
	%gives  $s_{\lambda/\mu}$; overall we get $n$ times $s_{\lambda/\mu}$.
	%}
	
	
	
	%\begin{proposition}\label{prop:QS_hook}
	%For every $0 \le k \le n-1$,
	%\[
	%s_{(n-k,1^k)} 
	%= \sum_{A \in \cnes\atop r(A)>k} (-1)^{r(A)-k-1} \hF^{\cyc}_{n,A}.
	%\]
	%Note that one of the summands is always
	%$\hF^{\cyc}_{n,\{[n]\}} \not\in \cQSym^-$.
	%\end{proposition}
	
	%\todo{RA: Is there a generalization to ribbon Schur functions?\\
	%YR: 
	%%Let $\alpha=(\alpha_1,\dots,\alpha_t)$ be a composition of $n$
	%%and $z_\alpha$ be a ribbon shape with row lengths $\alpha_1,\dots,\alpha_t$.
	%For a subset $J=\{j_1  < \ldots < j_t\} \subseteq [n-1]$, 
	%let $\alpha(J)$ be the composition of $n$ defined by
	%\[
	%\alpha(J):= (j_1,j_2-j_1,j_3-j_2,\ldots,j_t-j_{t-1},n-j_t).
	%\]
	%Let $z(J)$ be the ribbon with row lengths given by the entries of $\alpha(J)$
	%and let $M(J)$ be the horizontal strip with component sizes given by the entries of %$\alpha(J)$. By [reference ??], the associated skew ribbon Schur function satisfies
	%\[
	%ns_{z(J)}
	%= n\sum_{\varnothing \subseteq I \subseteq J}
	%(-1)^{\#(J \setminus I)} h_{\alpha(I)}= n\sum_{\varnothing \subseteq I \subseteq J}
	%(-1)^{\#(J \setminus I)} s_{M(I)}.
	%\]
	%By Theorem~\ref{conj:QS1} and Lemma~\ref{t:Fcyc_properties}, the RHS is equal to 
	%\[
	%\sum_{\varnothing \subsetneq I \subseteq J}
	%(-1)^{\#(J \setminus I)} \sum\limits_{T\in \SYT(M(I))} F^{\cyc}_{n,\cDes(T)}+(-1)^{\#(J)} F^{\cyc}_{n,\varnothing}.
	%\]
	%}
	
	
	%By Theorem~\ref{conj:QS1} and Proposition~\ref{prop:QS_hook},
	
	%\begin{corollary}
	%The ring of symmetric functions %lies in the subspace 
	%is a subring of $CQSym$.
	%% spanned by the (modified fundamental) cyclic quasi-symmetric functions. 
	%\end{corollary}
	
	%\todo{		It is easy to see that the  set $\{F^{\cyc}_{n, \tilde \alpha}:\ \tilde\alpha\ \text{is a cyclic coposition of $n$}\}$ is not independent.\\
	
	%	Note that it is not difficult to prove
	
	%\begin{obs}
	%The set $\{F^{\cyc}_{n, \tilde \alpha}:\ \tilde\alpha\neq (n)\ \text{is a cyclic coposition of $n$}\}$ %of cyclic quasi-symmetric functions
	%is an independent set in the ring of quasi-symmetric functions (considered as a vector space). 
	%\end{obs}
	
	%This statement would suffice for all shapes for which
	%$\cDes$ of a SYT could not be a singleton; namely, to all non-horizontal strips.\\
	
	%	Theorem~\ref{obs:QS1} suffices for applications proposed below, since $[n]$ does not appear as a $\cDes$ of any SYT.\\
	
	%}
	
	%We are now in a position to 
	%extend Corollary~\ref{t:nonnegative_structure_constants_minus} to $\cQSym$.
	
	
	
	%The {\em cyclic composition} which corresponds to a subset  $J=\{j_1<j_2<\cdots<j_t\}\subseteq [n]$ is
	%$\cc(J):=(j_2-j_1, j_3-j_2,\dots, j_t- j_{t-1}, j_1-j_t \mod n)$.
	%Recall the {\em cyclic ribbon Schur function}, defined as
	%\[
	%\cs_{\cc(J)} := \sum_{\varnothing \neq I \subseteq J}
	%(-1)^{\#(J \setminus I)} h_{\cc(I)} \in \Sym_n
	%\qquad (\forall \varnothing \ne J \subseteq [n]).
	%\]
	%%Furthermore,
	%The following cyclic analogue of~\cite[Theorem 3]{Gessel} holds.
	
	%\begin{lemma}\label{lem:f}
	%For every (homogeneous of degree $n$) symmetric function $f \in \Sym_n$, the unique expansion
	%\[
	%f = \sum_{A \in \cnes} c_A \hF^{\cyc}_{n,A}
	%\] 
	%satisfies, for each representative $J$ of each orbit $A \in \cnes$:
	%\[
	%c_A = \langle f, \tilde s_{\cc(J)} \rangle,
	%%\qquad (\forall A = [J] \in \cnes).
	%\]
	%where $\langle \cdot, \cdot \rangle$ is the usual inner product of symmetric functions.
	%\end{lemma}
	
	
	
	%\begin{corollary}\label{cor:integrality}
	%Let $f$ be a symmetric function, homogeneous of degree $n$.
	%The coefficients in the expansion of $f$ as a linear combination of %normalized fundamental cyclic quasi-symmetric functions 
	%$\left\{ \hF^{\cyc}_{n,A} \,:\, A \in \cnes \right\}$ 
	%are integral
	%if and only if the coefficients in its expansion as a linear combination of Schur functions are integral.
	%\end{corollary}
	
	Recall  {\em Postnikov's toric Schur functions} from~\cite{Postnikov}.
	
	%\begin{remark}
	\begin{proposition}\label{rem_Postnikov} 
		%By Lemma~\ref{lem:f} together with~\cite[Proposition 3.14]{ARR_CDes},  
		For 
		every non-hook shape $\lambda$, the coefficient of $\hF^{cyc}_{n,[J]}$ in $s_\lambda$ is equal to the coefficient of $s_\lambda$ in the Schur expansion of Postnikov's toric Schur function $s_{\mu(J)/1/\mu(J)}$.
		%, see \cite[Corollary 5.7]{ARR_CDes}. 
	\end{proposition}
	By~\cite[Theorem 5.3]{Postnikov} %it follows 
	%was proved by Postnikov 
	%that 
	these coefficients are equal to certain Gromov-Witten invariants. 
	%This requires an explanation
	%\end{remark}
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section{Enumerative applications}\label{sec:enumeration}
	
	%\subsection{Distribution of cyclic descents over 
	%shuffles of cyclic permutations}
	%cyclic shuffles}
	
	%\todo{In this subsection we follow Gessel's notes, with a slightly different definition of $\Psi$.}
	
	%In this section we present an application to distribution of cyclic descents over 
	%shuffles of cyclic permutations. More applications are presented in the full version paper.
	
	%\medskip
	
	%The  following classical result is due to Stanley. For bijective proofs see~\cite{Goulden, Stadler}.
	
	%\begin{proposition}[{\cite[Ex. 3.161]{EC1}}]\label{Stanley_lemma}
	%Given two permutations $\sigma$ and $\tau$ of disjoint sets of integers, 
	%the distribution of the descent set over all shuffles of $\sigma$ and $\tau$ depends only on $\Des(\sigma)$ and $\Des(\tau)$.
	%\end{proposition}
	
	%We have the following cyclic analogue. Recall that for every cyclic permutation
	%$[\DAG\pi]\in \symm_n/\ZZ_n$, $\cDes([\DAG \pi])$ is 
	%%a cyclic composition, 
	%defined up to cyclic rotation.
	
	\noindent
	\cref{thm:main1} implies the following analogue of 
	%Stanley's %well known
	the shuffling theorem~\cite[Ex.~3.161]{EC1} 
	%.[{\cite[Ex.~3.161]{EC1}}; see also {
	(see also~\cite[section 2.4]{GesselZhuang}).
	%}] 
	%for a combinatorial proof.]
	
	\begin{proposition}
		Let $A$ and $B$ be two disjoint sets of integers.
		For each  $u \in \symm_A$ and $v \in \symm_B$,
		the distribution of the cyclic descent set over all cyclic shuffles of $[\DAG u]$ and $[\DAG v]$ depends only on $\cDes([\DAG u])$ and $\cDes([\DAG v])$.
	\end{proposition}
	
	%\medskip
	
	Consider now $\ZZ[[q]]$, the ring of formal power series in $q$, as a (free abelian) additive group with generators $(q^n)_{n=0}^{\infty}$, and define a new product by
	\[
	q^i \odot q^j := q^{\max(i,j)},
	\]
	extended linearly.
	%\todo{Gessel's definition was:
	%\[
	% a_k q^k \odot \sum\limits_k b_k q^k:= \sum\limits_k k a_k b_k q^k.
	%\]
	%}
	We obtain a (commutative and associative) ring, to be denoted $\ZZ[[q]]_{\odot}$.
	
	Consider also the ring $\ZZ[[\xx]] = \ZZ[[x_1, x_2, \ldots]]$, and its subring $\ZZ[[\xx]]_\bded$ consisting of bounded-degree power series.  Define a map $\Psi: \ZZ[[\xx]]_\bded \to \ZZ[[q]]_{\odot}$ by
	\[
	\Psi(x_{i_1}^{m_1} \cdots x_{i_k}^{m_k}) := q^{i_k}
	\qquad (k > 0,\, i_1 < \cdots < i_k,\, m_1, \ldots, m_k > 0)
	\]
	and
	%\[
	%\Psi(1) := 1,
	%\]
	$\Psi(1) := 1$, extended linearly.
	
	\begin{obs}
		$\Psi$ is a ring ($\ZZ$-algebra) homomorphism.
	\end{obs}
	
	%We now compute the images, under $\Psi$, of various quasi-symmetric 
	%and cyclic quasi-symmetric 
	%functions. All explicit formulas will be stated in $\ZZ[[q]]$, with the standard multiplication.
	
	\begin{lemma}\label{t:Psi_cyc_M_and_F}
		For any positive integer $n$,
		%\[
		%\Psi(F_{n,J}) 
		%= \frac{q^{\,|J|+1}}{(1-q)^{n}}
		%= (1-q) \sum_r \binom{r+n-|J|-1}{n} q^r \qquad (\forall J\subseteq [n-1]).
		%\]
		%and
		\[
		\Psi(F^{\cyc}_{n,J}) 
		= \frac{|J| q^{|J|} + (n - |J|) q^{|J|+1}}{(1-q)^{n}}
		= (1-q) \sum_r \binom{r+n-|J|-1}{n-1} r q^r \qquad (\forall J\subseteq [n]).
		\]
	\end{lemma}
	%\todo{For Gessel's original $\Psi$,
	%\[
	%\Psi(F^{\cyc}_{n,J})
	%= \frac{q^{|J|}}{(1-q)^n}
	%= \sum_{r} \binom{n+r-|J|-1}{n-1} q^r.
	%\]
	%}
	
	
	%\begin{lemma}\label{t:Psi_cyc_MM_and_FF}
	%For any positive integers $m$, $n$,
	%\[
	%\Psi(F_{m,J} \cdot F_{n,K})
	%= (1-q) \sum_{r} \binom{r+m-|J|-1}{m} \binom{r+n-|K|-1}{n} q^r \qquad (\forall J \subseteq [m-1],\ K \subseteq [n-1]).
	%\]
	%and
	%\[
	%\Psi(F^{\cyc}_{m,J} \cdot F^{\cyc}_{n,K}) 
	%= (1-q) \sum_{r} \binom{r+m-|J|-1}{m-1} \binom{r+n-|K|-1}{n-1} r^2 q^r \qquad (\forall J \subseteq [m],\ K \subseteq [n]).
	%\]
	%\end{lemma}
	
	Using \cref{thm:main1} and \cref{t:Psi_cyc_M_and_F} we prove
	
	\begin{theorem}\label{t:des_of_shuffle}
		Let $A$ and $B$ be two disjoint sets of integers with  $|A|=m$ and $|B|=n$.
		For each  $u\in \symm_A$ and $v\in \symm_B$,
		%Let  $u$ and $v$ be two permutations on disjoint sets of
		%integers $U$ and $V$ respectively, with $|U|=m$ and $|V|=n$.
		%\begin{itemize}
		%\item[1.] The distribution of descent number over all shuffles of
		%$\sigma$ and $\tau$ is given by
		%\[
		%\sum_{w \in \sigma \shuffle \tau} q^{\des(w)}
		%= (1-q)^{m+n+1} \sum_{r} \binom{r+m-\des(\sigma)}{m} \binom{r+n-\des(\tau)}{n} q^r.
		%\]
		%\item[2.]
		the distribution of the cyclic descent number over all cyclic shuffles of
		$[\DAG u]$ and $[\DAG v]$ is given by
		\[
		\sum_{[\DAG{w}] \in [\DAG{u}] \shuffle_{\cyc} [\DAG{v}]} q^{\cdes(w)}
		= (1-q)^{m+n} \sum_r \binom{r+m-\cdes(u)-1}{m-1} \binom{r+n-\cdes(v)-1}{n-1} r q^r.
		\]
		%\end{itemize}
	\end{theorem}
	
	
	\cref{t:des_of_shuffle} %together with certain binomial identities 
	implies \cref{thm:main3}. % and its following non-cyclic analogue.
	For other applications see the full version~\cite{AGRR}.
	
	%\begin{corollary}\label{t:num_shuffles_given_des}
	%Let  $\sigma$ and $\tau$ be two permutations on disjoint sets of
	%integers $U$ and $V$ respectively, with $|U|=m$ and $|V|=n$. 
	%%\begin{itemize}
	%%\item[1.]
	%If $\des(\sigma) = i$ and $\des(\tau) = j$ then the number of shuffles of $\sigma$ and $\tau$ with descent number $k$ is equal to
	%\[
	%\binom{m+j-i}{k-i} \binom{n+i-j}{k-j}.
	%\]
	%%\item[2.] 
	%%If $\cdes(\sigma) = i$ and $\cdes(\tau) = j$ then the number of cyclic shuffles of %%$[\DAG\sigma]$ and $[\DAG\tau]$ with cyclic descent number $k$ is equal to
	%%\[
	%%%\begin{aligned}
	%%a(m,n,i,j,k) 
	%%%&
	%%:= 
	%%k \binom{m+j-i-1}{k-i} \binom{n+i-j-1}{k-j}
	%%+ (m+n-k) \binom{m+j-i-1}{k-i-1} \binom{n+i-j-1}{k-j-1} \\
	%%&\;
	%%= 
	%\frac{k(m-i)(n-j) + (m+n-k)ij}{(m+j-k)(n+i-k)} \binom{m+j-i-1}{k-i} \binom{n+i-j-1}{k-j}. 
	%%\end{aligned}
	%%:= (m-i) \binom{m+j-i-1}{k-i-1} \binom{n+i-j}{k-j} + i \binom{m+j-i}%{k-i} %\binom{n+i-j-1}{k-j}.
	%\]
	%\[
	%=(m-i) \binom{m+j-i-1}{k-i-1} \binom{n+i-j}{k-j} + i \binom{m+j-i}{k-i} \binom{n+i-j-1}{k-j}.
	%\]
	%\end{itemize}
	%\end{corollary}
	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
	
	\section{Open problems and final remarks}\label{sec:open_problems}
	
	%\subsection{Enumeration}\ \\
	
	%We close with several remarks and questions.
	
	%\medskip
	
	%Garsia and Wallach~\cite{GW} proved that
	%the ring $\QSym_\QQ[x_1,\dots,x_n]$ of quasi-symmetric functions in $n$ variables (over $\QQ$) is a free module over the ring $\Sym_\QQ[x_1,\dots,x_n]$ of symmetric functions in these variables.
	%They also studied the quotient of $\QSym_\QQ[x_1,\dots,x_n]$ by the ideal generated by homogeneous symmetric functions of positive degree, proved that it has dimension $n!$ and computed its Hilbert series. 
	%Lauve and Mason~\cite{LauveMason} found an explicit basis.
	%Aval, Bergeron and Bergeron~\cite{ABB} studied analogous questions concerning the quotient of the ring $\QQ[x_1,\ldots,x_n]$ by the ideal generated by homogeneous quasi-symmetric functions. % of positive degree.
	%%%%%%
	%Observing that
	%\[
	%\Sym_{\QQ}[x_1,\ldots,x_n] \subseteq \cQSym_{\QQ}[x_1,\ldots,x_n] \subseteq %\QSym_{\QQ}[x_1,\ldots,x_n] \subseteq \QQ[x_1,\ldots,x_n],
	%\]
	%it is desired to solve the following problem.
	
	%\begin{problem} %\ \\
	%%\begin{itemize}
	%%\item[1.] 
	%Are the following modules free? \ (i) $\cQSym_{\QQ}[x_1,\ldots,x_n]$ over $\Sym_{\QQ}[x_1,\ldots,x_n]$,\ (ii) $\QQ[x_1,\ldots,x_n]$ over $\cQSym_{\QQ}[x_1,\ldots,x_n]$,\ and (iii) $\QSym_{\QQ}[x_1,\ldots,x_n]$ over $\cQSym_{\QQ}[x_1,\ldots,x_n]$.
	%\begin{itemize}
	%\item[(i)] 
	%$\cQSym_{\QQ}[x_1,\ldots,x_n]$ over $\Sym_{\QQ}[x_1,\ldots,x_n]$,
	%the ideal generated by homogeneous symmetric functions of positive degree;
	%\item[(ii)] 
	%$\QQ[x_1,\ldots,x_n]$ over $\cQSym_{\QQ}[x_1,\ldots,x_n]$,
	%the ideal generated by homogeneous cyclic quasi-symmetric functions of positive degree; 
	%and 
	%\item[(iii)] 
	%$\QSym_{\QQ}[x_1,\ldots,x_n]$ over $\cQSym_{\QQ}[x_1,\ldots,x_n]$.
	%the ideal generated by homogeneous cyclic quasi-symmetric functions of positive degree.
	%\end{itemize}
	%\item[2.] 
	%In case of affirmative answers, compute the Hilbert series. 
	%of the quotients by the corresponding ideals.
	%\end{itemize}
	%\end{problem}
	
	
	A Schur-positivity phenomenon, involving cyclic quasi-symmetric functions, was presented in \cref{sec:expansion}. It is desired to find more results of this type. 
	%Here is a conjectured one.
	For example, it was proved in~\cite[Cor. 7.7]{ER3} that, 
	for any $0< k< n$,
	the cyclic quasi-symmetric function
	\[
	%\sum_{\substack{\pi \in \symm_n \\ \cdes(\pi^{-1}) = k}} F_{n,\Des(\pi)}
	\sum_{\pi \in \symm_n \,:\, \cdes(\pi^{-1}) = k} F_{n,\Des(\pi)}
	\]
	is symmetric and Schur-positive.
	Computational experiments suggest the following refined cyclic version.
	
	%Another problem deals with Schur-positivity phenomenon.
	
	%There exists a very extensive literature regarding Schur-positivity of %phenomena which involve
	%various quasi-symmetric functions. %, see e.g.~\cite{Gessel-Reutenauer} and follow ups.
	%A Schur-positivity phenomenon, involving cyclic quasi-symmetric functions, was presented in Section~\ref{sec:expansion}. It is desired to find more results of this type. Here is a conjectured one.
	
	%It was proved in~\cite[Cor. 7.7]{ER3} that,  for every positive integers $0< k< n$, 
	%the cyclic quasi-symmetric function
	%$
	%\sum_{\pi \in \symm_n : \cdes(\pi^{-1}) = k} F_{n,\Des(\pi)}
	%$
	%is symmetric and Schur-positive.
	%Computational experiments suggest the following 
	%(stronger) cyclic version.
	%Schur-positivity phenomenon.
	%holds.
	%that sum of fundamental cyclic QSF over inverse cyclic shuffles is Schur positive. Namely,
	
	%Computational experiments suggest the following 
	%Schur-positivity phenomenon.
	
	\begin{conjecture} 
		For every $\varnothing \subsetneq J \subsetneq [n]$ the cyclic quasi-symmetric function
		\[
		\sum_{\substack{\pi \in \symm_n \\ [\cDes(\pi^{-1})] = [J]}} F^\cyc_{n,\cDes(\pi)}
		= \sum_{\substack{\pi\in \symm_n \\ (\exists i)\ \cDes(\pi^{-1}) = J+i}} F^\cyc_{n,\cDes(\pi)}
		\]
		is symmetric and Schur-positive.
	\end{conjecture}
	
	%\medskip
	
	%Finally, c
	Cyclic descents were introduced by Cellini~\cite{Cellini} in the search for subalgebras of Solomon's descent algebra. An important subalgebra of the descent algebra is the peak algebra.
	
	\begin{problem}
		Define and study cyclic peaks and a cyclic peak algebra.
	\end{problem}
	
	%For cyclic peaks in Dick paths see~\cite[\S 5.1]{AER} .
	
	%\acknowledgements{???}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%% if you use biblatex then this generates the bibliography
	%% if you use some other method then remove this and do it your own way
	%\printbibliography
	
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%		
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