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\topmatter
\title Simultaneous maj statistics
\endtitle

\author{Dongsu Kim and Dennis Stanton}
\endauthor
\thanks
The first author is partially supported by KOSEF: 971--0106--038--2.
\endthanks
\address
Department of Mathematics, KAIST, Taejon 305--701, Korea
\newline\indent
School of Mathematics, University of Minnesota,
Minneapolis, MN 55455
\endaddress

\email dskim\@math.kaist.ac.kr stanton\@math.umn.edu
\endemail
\abstract{The generating function for words with several simultaneous
$maj$ weights is given. New $maj$-like Mahonian statistics result. Some
applications to integer partitions are given.}
\endabstract

\endtopmatter
\document

{\subheading{1. Introduction}}

The usual $maj$ statistic \cite{2} on words $w$
is defined by adding the location of the
descents of the word $w$,
$$
maj(w)=\sum_{i: w_i>w_{i+1}} i.
$$
This definition presumes that the alphabet for the letters of $w$ have been
linearly ordered, for example $2>1>0$,
$$
maj(1102201)=2+5=7=maj_{210}(1102201).
$$
However a similar definition can be made assuming any linear ordering $\sigma$;
here we take $1>2>0$, $\sigma=120$, and $2>0>1$, $\sigma=201$
$$
maj_{120}(1102201)=2+5=7, \quad maj_{201}(1102201)=5+6=11.
$$

In this paper we consider the generating function for several such simultaneous
$maj$ statistics (see Corollary 1). A more general generating function is given
(Theorem 3), and some applications to Mahonian statistics (Corollary 2) and
integer partitions (Theorem 4) are stated.

We first give a 3 letter theorem, which motivates the general result
(Theorem~3). Let $W(m,n,k)$ be the set of words of length $m+n+k$ with $m$~0's,
$n$~1's and $k$~2's.

\proclaim{Theorem 1} For any non-negative integers $m$, $n$, and $k$ we have
$$
\align
\sum_{w\in W(m,n,k)} &x^{maj_{120}(w)}y^{maj_{201}(w)}z^{maj_{012}(w)}
=x^{n+k}y^k\qtri{m+n+k-1}{m-1}{n}{k}{xyz}+\\
&
y^{k+m}z^m\qtri{m+n+k-1}{m}{n-1}{k}{xyz}+
z^{m+n}x^n\qtri{m+n+k-1}{m}{n}{k-1}{xyz}.
\endalign
$$
\endproclaim

\demo{Proof} We prove a stronger statement, that the three terms in Theorem~1
are the generating functions for the words in $W(m,n,k)$ ending in 0, 1, and 2
respectively.

We proceed by induction on $m+n+k$. If $w$ ends in a $0$, the penultimate
letter must be either $0$, $1$ or $2$. Using induction we must verify that
$$
\align
x^{n+k}y^k&\qtri{m+n+k-1}{m-1}{n}{k}{xyz}
=x^{n+k}y^k\qtri{m+n+k-2}{m-2}{n}{k}{xyz}+\\
&x^{m+n+k-1}y^{m+k-1}z^{m-1}\qtri{m+n+k-2}{m-1}{n-1}{k}{xyz}+\\
&(xy)^{m+n+k-1}z^{m+n-1}x^n\qtri{m+n+k-2}{m-1}{n}{k-1}{xyz},
\endalign
$$
which is the well-known recurrence formula \cite{1} for the $xyz$-trinomial
coefficient.

The other two cases are verified similarly.
\qed\enddemo

It should be noted that if any two of $x,y,z$ are set equal to $1$, then the
usual $maj$ generating function as a $q$-trinomial coefficient results.


{\subheading{2. A 7-variable theorem}}

Theorem 1 contains three free variables, $x,y$ and $z$. In this section we
generalize Theorem 1 to Theorem 2, which contains seven free variables. Then we
indicate how to specialize Theorem 2 to obtain new explicit classes of Mahonian
statistics on words of $0$'s, $1$'s, and $2$'s.

Suppose that the weights of the various possible ascents and descents in
position $m+n+k-1$ of a word $w$ of $m$ 0's, $n$ 1's, and $k$ 2's are given by
\roster
\item"(wt10)" $a_0^{m-1}a_1^{n}a_2^k$ for a descent $10$,
\item"(wt21)" $b_0^mb_1^{n-1}b_2^{k}$ for a descent $21$,
\item"(wt20)" $c_0^{m-1}c_1^{n}c_2^k$ for a descent $20$,
\item"(wt01)" $d_0^md_1^{n-1}d_2^{k}$ for an ascent $01$,
\item"(wt12)" $e_0^{m}e_1^ne_2^{k-1}$ for an ascent $12$,
\item"(wt02)" $f_0^{m}f_1^nf_2^{k-1}$ for an ascent $02$.
\endroster
Also suppose that the generating function for all such words $w$ has the form
$$
\align
p_0(n,k)\!\qtri{\!m+n+k-1\!}{m-1}{n}{k}{B}\!\!+
&p_1(k,m)\!\qtri{\!m+n+k-1\!}{m}{n-1}{k}{B}\!\!\\
&+p_2(m,n)\!\qtri{\!m+n+k-1\!}{m}{n}{k-1}{B}
\tag2.1
\endalign
$$
for some base $B$, and $p_0(n,k)=p_{01}^np_{02}^k$,
$p_1(k,m)=p_{11}^kp_{12}^m$, $p_2(m,n)=p_{21}^mp_{22}^n$. We also assume that
the three terms in (2.1) correspond to the $w$ which end in $0$, $1$, and $2$
respectively.

Thus we have 25 free variables
$$
\cup_{i=0}^2\{a_i,b_i,c_i,d_i,e_i,f_i,p_{i1},p_{i2}\}\cup\{B\}.
$$

These 25 variables are related by the three equations which we require by
induction
$$
\align
p_0(n,k)\qtri{m+n+k-1}{m-1}{n}{k}{B}=p_0(n,k)&\qtri{m+n+k-2}{m-2}{n}{k}{B}\\
+a_0^{m-1}a_1^{n}a_2^k\ p_1(k,m-1)&\qtri{m+n+k-2}{m-1}{n-1}{k}{B}\\
+c_0^{m-1}c_1^{n}c_2^k\ p_2(m-1,n)&\qtri{m+n+k-2}{m-1}{n}{k-1}{B},
\tag2.2a
\endalign
$$
$$
\align
p_1(k,m)\qtri{m+n+k-1}{m}{n-1}{k}{B}=p_1(k,m)&\qtri{m+n+k-2}{m}{n-2}{k}{B}\\
+b_0^{m}b_1^{n-1}b_2^k\ p_2(m,n-1)&\qtri{m+n+k-2}{m}{n-1}{k-1}{B}
\\
+d_0^{m}d_1^{n-1}d_2^k\ p_0(n-1,k)&\qtri{m+n+k-2}{m-1}{n-1}{k}{B},
\tag2.2b
\endalign
$$
$$
\align
p_2(m,n)\qtri{m+n+k-1}{m}{n}{k-1}{B}=p_2(m,n)&\qtri{m+n+k-2}{m}{n}{k-2}{B}\\
+f_0^{m}f_1^{n}f_2^{k-1}\ p_0(n,k-1)&\qtri{m+n+k-2}{m-1}{n}{k-1}{B}\\
+e_0^{m}e_1^{n}e_2^{k-1}\ p_1(k-1,m)&\qtri{m+n+k-2}{m}{n-1}{k-1}{B}.
\tag2.2c
\endalign
$$
We do not know the general solution to the equations (2.2a-c). However, we will
give the general solution to (2.2a-c) if we make another assumption. If we
specify that the coefficient of the second term on the the right side of (2.2a)
is $B^{m-1}$ times the coefficient of the first term, and the coefficient of
the third term is $B^{m+n-1}$ times the coefficient of the first term, then the
$B$-trinomial recurrence relation verifies (2.2a). These two equations are
$$
\aligned
a_0^{m-1}a_1^na_2^kp_{11}^kp_{12}^{m-1}=&B^{m-1}p_{01}^n p_{02}^k,\\
c_0^{m-1}c_1^nc_2^kp_{21}^{m-1}p_{22}^{n}=&B^{m+n-1}p_{01}^n p_{02}^k.
\endaligned
\tag2.3a
$$
Similarly, we assume the $B$-trinomial recurrence for (2.2b) and (2.2c), which
become
$$
\aligned
b_0^{m}b_1^{n-1}b_2^kp_{21}^mp_{22}^{n-1}=&B^{n-1}p_{11}^k p_{12}^m,\\
d_0^{m}d_1^{n-1}d_2^kp_{01}^{n-1}p_{02}^{k}=&B^{n+k-1}p_{11}^k p_{12}^m.
\endaligned
\tag2.3b
$$
and
$$
\aligned
f_0^{m}f_1^{n}f_2^{k-1}p_{01}^np_{02}^{k-1}=&B^{k-1}p_{21}^m p_{22}^n,\\
e_0^{m}e_1^{n}e_2^{k-1}p_{11}^{k-1}p_{12}^{m}=&B^{k+m-1}p_{21}^m p_{22}^n.
\endaligned
\tag2.3c
$$

Since these equations should hold for all $m$, $n$ and $k$, each of these 6
equations contains 3 equations (one each in $m$, $n$, and $k$). Thus we have 18
equations in the 25 free variables, which are written in a matrix form, where
the first column comes from the equations in (2.3a):
$$
\pmatrix
p_{12}a_0 &p_{21}b_0 &f_0       \\
a_1       &p_{22}b_1 &p_{01}f_1 \\
p_{11}a_2 &b_2       &p_{02}f_2 \\
p_{21}c_0 &d_0       &p_{12}e_0 \\
p_{22}c_1 &p_{01}d_1 &e_1       \\
c_2       &p_{02}d_2 &p_{11}e_2
\endpmatrix
=
\pmatrix
B      &p_{12} &p_{21} \\
p_{01} &B      &p_{22} \\
p_{02} &p_{11} &B      \\
B      &p_{12} &p_{21}B\\
p_{01}B&B      &p_{22} \\
p_{02} &p_{11}B&B
\endpmatrix .
$$

One may find the general solution to these 18 equations, leaving 7 free
variables
$$
\{a_0,a_1,a_2,b_0,b_1,b_2,B\}.
$$
The explicit solutions for the remaining 18 variables are given below. The
weights (wt) become (W):
\roster
\item"(W10)" $a_0^{m-1}a_1^{n}a_2^k$ \quad for a descent $10$,
\item"(W21)" $b_0^mb_1^{n-1}b_2^k$ \quad for a descent $21$,
\item"(W20)" $(a_0b_0)^{m-1}(a_1b_1)^{n}(a_2b_2)^{k}$ \quad for a descent $20$,
\item"(W01)" $(B/a_0)^{m}(B/a_1)^{n-1}(B/a_2)^{k}$ \quad for an ascent $01$,
\item"(W12)" $(B/b_0)^{m}(B/b_1)^n(B/b_2)^{k-1}$ \quad for an ascent $12$,
\item"(W02)" $(B/a_0b_0)^{m}(B/a_1b_1)^n(B/a_2b_2)^{k-1}$ \quad for an
ascent $02$,
\endroster
and
$$
\align
p_0(n,k)&=a_1^n(a_2b_2)^k, \quad p_1(k,m)=b_2^k(B/a_0)^m,\\
 p_2(m,n)&=(B/a_0b_0)^m(B/b_1)^n.
\endalign
$$

\proclaim{Theorem 2} The generating function of all words $w\in W(m,n,k)$
with weights given by (W) is
$$
\align
a_1^n(a_2b_2)^k &\qtri{m+n+k-1}{m-1}{n}{k}{B}+
b_2^k(B/a_0)^m \qtri{m+n+k-1}{m}{n-1}{k}{B}+\\
&(B/a_0b_0)^m(B/b_1)^n \qtri{m+n+k-1}{m}{n}{k-1}{B}.
\endalign
$$
\endproclaim

Theorem 1 is the special case of Theorem 2 for which $B=xyz$,
\newline $a_0=a_1=a_2=x$, and $b_0=b_1=b_2=y$  hold.

There are 7 other versions of Theorem 2. These 8 theorems arise by
independently replacing the pair of factors $(B^{m-1}, B^{m+n-1})$ by
$(B^{m+k-1}, B^{m-1})$ in equation (2.3a), $(B^{n-1}, B^{n+k-1})$ by
$(B^{n+m-1}, B^{n-1})$ in equation (2.3b), and $(B^{k-1}, B^{k+m-1})$ by
$(B^{k+n-1}, B^{k-1})$ in (2.3c). The $B$-trinomial recurrence still holds. For
instance if we make a replacement in (2.3a),

$$
\aligned
a_0^{m-1}a_1^na_2^kp_{11}^kp_{12}^{m-1}=&B^{m+k-1}p_{01}^n p_{02}^k,\\
c_0^{m-1}c_1^nc_2^kp_{21}^{m-1}p_{22}^{n}=&B^{m-1}p_{01}^n p_{02}^k,
\endaligned
\tag2.3a$'$
$$
then the explicit solutions to (2.3a$'$) and (2.3b-c) give the weight (W$'$):
\roster
\item"(W$'$10)" $a_0^{m-1}a_1^{n}a_2^k$ \quad for a descent $10$,
\item"(W$'$21)" $b_0^mb_1^{n-1}b_2^k$ \quad for a descent $21$,
\item"(W$'$20)" $(a_0b_0)^{m-1}(a_1b_1/B)^{n}(a_2b_2/B)^{k}$ \quad for
a descent $20$,
\item"(W$'$01)" $(B/a_0)^{m}(B/a_1)^{n-1}(B^2/a_2)^{k}$ \quad for an ascent
$01$,
\item"(W$'$12)" $(B/b_0)^{m}(B/b_1)^n(B/b_2)^{k-1}$ \quad for an ascent $12$,
\item"(W$'$02)" $(B/a_0b_0)^{m}(B/a_1b_1)^n(B^2/a_2b_2)^{k-1}$ \quad for an
ascent $02$,
\endroster
and the corresponding theorem is the following:
\proclaim{Theorem 2$'$} The generating function of all words $w\in W(m,n,k)$
with weights given by (W$'$) is
$$
\align
a_1^n(a_2b_2/B)^k &\qtri{m+n+k-1}{m-1}{n}{k}{B}+
b_2^k(B/a_0)^m \qtri{m+n+k-1}{m}{n-1}{k}{B}+\\
&(B/a_0b_0)^m(B/b_1)^n \qtri{m+n+k-1}{m}{n}{k-1}{B}.
\endalign
$$
\endproclaim
We do not state the remaining 6 variations here.

We can find Mahonian statistics by requiring that the generating function in
Theorem~2 is the $B$-trinomial via the $B$-trinomial recurrence. There are six
choices for this recurrence, one for each ordering of the 3 terms. So Theorem~2
gives a total of $6$ possible Mahonian statistics, one of which
($maj_{012}$), is found by setting $a_0=a_1=a_2=b_0=b_1=b_2=1$.
Theorem~2$'$ also gives a total of $6$ possible Mahonian statistics, one of
which is found by setting $a_0=a_1=b_0=b_1=b_2=1,\ a_2=B$. Similarly there are
$6$ possible Mahonian statistics for each of other 6 versions of Theorem~2, for
a total of $6\!\times8=48$. Six of them are the six possible $maj_{\sigma}$
statistics, the remaining 42 come in 7 classes of six each, and they are all
variations on $maj$. Each class of size 6 consists of a $maj$ variation, and 5
others which correspond to 5 non-trivial reorderings of $\{0,1,2\}$ of that
$maj$ variation. We give below one member of each class, eight in total.

We start with an example from Theorem~2$'$.
If we set $a_0=a_1=b_0=b_1=b_2=1,\ a_2=B$ in
Theorem~2$'$, the weight (W$'$) reduces to
\roster
\item"(W$'$10)" $B^k$ \quad for a descent $10$,
\item"(W$'$21)" 1 \quad for a descent $21$,
\item"(W$'$20)" $B^{-n}$ \quad for a descent $20$,
\item"(W$'$01)" $B^{m+n+k-1}$ \quad for an ascent $01$,
\item"(W$'$12)" $B^{m+n+k-1}$ \quad for an ascent $12$,
\item"(W$'$02)" $B^{m+n+k-1}$ \quad for an ascent $02$.
\endroster
Note that the above weight (W$'$) is a perturbation of
$maj_{012}$ involving the descents $10$ and $20$. We write it
as  $maj_{012}+s_0$, where $s_0$ is defined in the following way.
We define $s_0$ by giving the non-zero values at adjacent
letters. One then adds these values to find $s_0$. It is assumed that
if $w$ is truncated after the adjacent letters, $w$ has $m$ 0's, $n$ 1's,
and~$k$~2's.

\flushpar
$s_0(w)$:
\roster
\item ${k}$ \quad for an adjacent $10$,
\item ${-n}$ \quad for an adjacent $20$.
\endroster
For example,
$$
s_0(22012110201)=-0+3-3=0.
$$

It turns out (we do not write
down the details here) that the eight statistics (including $maj_{012}$) can be
defined by three independent perturbations of $maj_{012}$: $s_0$, $s_1$, and
$s_2$. For any subset $A\subset\{0,1,2\}$ put
$$
s_A(w)=\sum_{i\in A} s_i(w).
$$
Then the eight Mahonian statistics are $maj_{012}+s_A$. In fact the set $A$
indicates which replacements are made in (2.3a-c). For instance the above
(W$'$) is $maj_{012}+s_{\{0\}}$ and if we make replacements, say in (2.3b) and
(2.3c), then the corresponding statistics will be $maj_{012}+s_{\{1,2\}}$, and
so on. We define $s_1$, $s_2$ analogously by giving the non-zero values at
adjacent letters. One then adds these values to find the statistic. It is
assumed that if $w$ is truncated after the adjacent letters, $w$ has $m$ 0's,
$n$ 1's, and~$k$~2's.

\flushpar
$s_1(w)$:
\roster
\item ${m}$ \quad for an adjacent $21$,
\item ${-k}$ \quad for an adjacent $01$.
\endroster
$s_2(w)$:
\roster
\item ${n}$ \quad for an adjacent $02$,
\item ${-m}$ \quad for an adjacent $12$.
\endroster

For example,
$$
s_1(22012110201)=-2+1-4=-5, \quad s_2(22012110201)=-1+3=2.
$$

Below is a table evaluating $maj_{012}$, $s_0$, $s_1$, and $s_2$ at the 6
permutations of $012$. Note that the $maj_{012}$ generating function is
$1+2B+2B^2+B^3$, which is also the generating function for $maj_{012}+s_A$, for
any subset $A\subset \{0,1,2\}$.
$$
\matrix
\text{word} &maj_{012} &\hfill s_0 &\hfill s_1 &\hfill s_2\\
012&3 &\hfill 0 &\hfill 0 &-1\\
021&1 &\hfill 0 &\hfill 1 &\hfill 0\\
102&2 &\hfill 0 &\hfill 0 &\hfill 1\\
120&1 &-1       &\hfill 0 &\hfill 0\\
201&2 &\hfill 0 &-1       &\hfill 0\\
210&0 &\hfill 1 &\hfill 0 &\hfill 0
\endmatrix
$$

We repeat that all 48 Mahonian statistics may be found from these 8 by
permuting the letters $0$, $1$, and $2$. In this case $maj_{012}$ becomes
$maj_{\sigma}$, and each $s_i$ is found by applying $\sigma$ to $0$, $1$, and
$2$ in the definition of $s_i$.

{\subheading{3. $N$ letters}}

In this section we briefly generalize Theorem 2 to words with $N$ letters in
Theorem 3. We state the $N$ letter version of Theorem 1 in Corollary 1. There
are $N!\,2^N$ Mahonian statistics, which come in $2^N$ families each of size
$N!$. We explicitly give the corresponding $2^N$ Mahonian statistics in
Corollary 2.

Let $W(a_0,a_1,\cdots, a_{N-1})$ be the set of all words $w$ with $a_i$ $i$'s,
$0\le i\le N-1$.

If the words $w$ have $N$ letters instead of 3 letters, then each adjacent pair
$ij$, $i\ne j$, could be weighted by $N$ variables, instead of 3 variables.
Also the coefficients $p_i$, $0\le i\le N-1$ would have $N-1$ variables.
Together with the base $B$, we have a total of $N(N^2-N)+N(N-1)+1=N^3-N+1$
variables. Each of the $N$ recurrences  required by induction gives $N(N-1)$
equations in these variables. So $N(N-1)+1$ variables will be free in the
multivariable version of Theorem 2.

In order to fully describe the resulting theorem, some care must be taken with
notation.

The $N(N-1)+1$ free variables may be taken to be the base $B$ along with the
$N$ weights of the adjacent pairs $(i+1)i$, for $i=0, \cdots, N-2$, for which
we use the variables
$$
(x_{i0},x_{i1},\cdots,x_{iN-1}), \quad 0\le i\le N-2.
$$

Suppose that $w$ ends in an adjacent pair $ij$, $i\ne j$, and that there are
$n_k$ $k$'s preceding the last letter $j$ of $w$. The weight of the pair $ij$
is given by
$$
\aligned
&\prod_{k=0}^{N-1} \bigl(\prod_{l=j}^{i-1}x_{lk}\bigr)^{n_k}
\quad\quad{\text{    if  }}j<i,\\
&\prod_{k=0}^{N-1} \bigl(B/\prod_{l=i}^{j-1}x_{lk}\bigr)^{n_k}
\quad{\text{  if  }}i<j.\\
\endaligned
\tag4.2
$$
As usual, we multiply the weights of adjacent pairs to find the weight of the
word~$w$.


\proclaim{Theorem 3} The generating function of all words
$w\in W(a_0,a_1,\cdots, a_{N-1})$ with weights given by (4.2) is
$$
\sum_{i=0}^{N-1}p_i(a_0,a_1,\cdots, a_{N-1})
\qbin{a_0+\cdots+a_{N-1}-1}{a_0,\cdots, a_i-1,\cdots,a_{N-1}}{B}
$$
where
$$
p_i(a_0,a_1,\cdots, a_{N-1})=\biggl(\prod_{l=0}^{i-1}
(B/\prod_{k=1}^{i-l}x_{i-k,l})^{a_l}
\biggr)\biggl(\prod_{l=i+1}^{N-1}(\prod_{k=0}^{l-i-1}x_{i+k,l})^{a_l}\biggr).
$$
\endproclaim

Note that $p_i$ in Theorem 3 is independent of $a_i$.

The multivariable version of Theorem 1 occurs if
$$
x_{i0}=x_{i1}=\cdots=x_{iN-1}=x_i, \quad 0\le i\le N-2,
$$
and $B=x_0x_1\cdots x_{N-1}$. Then the weights (4.2) become
$$
\align
&(x_j\cdots x_{i-1})^{n_0+\cdots+n_{N-1}} \qquad\qquad\quad{\text{ if }} j<i,\\
&(x_0\cdots x_{i-1} x_j\cdots x_{N-1})^{n_0+\cdots+n_{N-1}} \quad{\text{ if
}} i<j,
\endalign
$$
and the next corollary holds.

\proclaim{Corollary 1} We have
$$
\align
&\sum_{w\in W(a_0,\cdots,a_{N-1})}
\prod_{i=0}^{N-1} x_i^{maj_{i+1\cdots (N-1) 01\cdots i}(w)}=\\
&\sum_{i=0}^{N-1}p_i(a_0,a_1,\cdots, a_{N-1})
\qbin{a_0+\cdots+a_{N-1}-1}{a_0,\cdots, a_i-1,\cdots,a_{N-1}}{x_0\cdots
x_{N-1}}
\endalign
$$
where
$$
p_i(a_0,a_1,\cdots, a_{N-1})=\!\biggl(\prod_{l=0}^{i-1}
(x_0\cdots x_{l-1}x_i\cdots x_{N-1})^{a_l}\!
\biggr)\!
\biggl(\prod_{l=i+1}^{N-1}(x_i\cdots x_{l-1})^{a_l}\!\biggr).
$$
\endproclaim

We next give the $2^N$ Mahonian statistics which follow from Theorem 3. Again
they may be classified by perturbations of $maj_{01\cdots N-1}$. For any subset
$A\subset\{0,1,\cdots,N-1\}$, define
$$
s_A(w)=\sum_{i\in A} s_i(w).
$$
The individual statistics $s_i(w)$ only depend upon the subwords of $w$ ending
in $i$, as in \S2. For any given $i\in w$, suppose that $i$ is preceded by
$n_j$ $j$'s, $0\le j \le N-1$. Extend the definition of $n_j$ to be periodic
$\!\!\!\mod N$: $n_{j+N}=n_j$ for all $j$. If the letter preceding $i$ is
$i+k$, the contribution to $s_i(w)$ is positive on the circular interval
$[i+k+1,i-1]$ and negative on the circular interval $[i+1,i+k-1]$,
$$
(n_{i+k+1}+n_{i+k+2}+\cdots+n_{(i-1)})-
(n_{i+1}+n_{i+2}+\cdots+n_{i+k-1}).
\tag3.1
$$

We add the contributions of (3.1) over all $i\in w$ to find $s_i(w)$. There is
no contribution if $k=0$; that is, for a repeated $ii$. For example,
$$
s_1(41241012411312301)=0+(-1)+(-3)+(1-2)+(4-2)+(-8)=-11.
$$

\proclaim{Corollary 2} For any set $A\subset\{0,1,\cdots,N-1\}$,
the  statistic $maj_{01\cdots N-1}+s_A$ is Mahonian on $W(a_0,a_1,\cdots,
a_{N-1})$.
\endproclaim

These Mahonian statistics are examples of {\it{splittable}} statistics
\cite{3}.

One may also allow weights on the adjacent letters $00$, $11$, and $22$ for a
more general version of Theorem~3.

{\subheading{4. Applications to partitions}}

In this section we apply Theorem 1 and Theorem 3 to integer partitions.

The special case $k=0$, $z=1$, $x=y=q$ of Theorem 1 is
$$
\sum_{w\in W(m,n,0)} q^{maj_{10}(w)+maj_{01}(w)}=
\qbin{m+n}{m}{q^2}\frac{q^m+q^n}{1+q^{m+n}}:=f(m,n,q).
\tag4.1
$$
MacMahon \cite{4, p. 139} previously gave (4.1).

The following generating function (using standard notation found in \cite{1})
follows from (4.1),
$$
\sum_{m,n\ge 0} f(m,n,q)\frac{(xq)^m(yq)^n}{(q;q)_{m+n}}=
\frac{(xyq^2;q^2)_\infty}{(xq,yq;q)_\infty}.
\tag4.2
$$
One way to see (4.2) is to consider the generating function for pairs of
partitions $(\lambda,\mu)$ with distinct parts, weighted by
$$
x^{{\text{\# of parts of }} \lambda} y^{{\text{\# of parts of }}\mu}
q^{|\lambda|+|\mu|}
$$
which is
$$
\prod_{k=1}^\infty \left(1+\frac{xq^k}{1-xq^k}+\frac{yq^k}{1-yq^k}\right)=
\frac{(xyq^2;q^2)_\infty}{(xq,yq;q)_\infty}.
$$

To prove (4.1), we must find a weight preserving bijection $\phi$ from the set
of such $(\lambda,\mu)$, \# parts of $\lambda =m$, \# parts of $\mu =n$, to the
set of ordered pairs $(w,\gamma)$, where $w\in W(m,n,0)$, and $\gamma$ is a
partition with $m+n$ parts.

To define $w$, order the $m+n$ parts of $\lambda \cup \mu$ into a partition
$\theta$, and let $w_i=0$ if $\theta_i\in \lambda$, $w_i=1$ if $\theta_i\in
\mu$. This is well defined since the parts of $\lambda$ and $\mu$ are distinct.
To define $\gamma$, let $t_i$ be the number of descents
or ascents to the right of position $i$ in the word $w$.
Then we let $\gamma=\theta-t$. For example if
$$
\lambda=7742,\quad \mu=88661,
$$
then
$$
\theta=887766421,\qquad w=110011001,\quad t=443322110,  \qquad
\gamma=444444311.
$$
This correspondence is the desired bijection $\phi$.

The natural analog of $\phi$ on triples $(\lambda,\mu,\theta)$ without pairwise
common parts produces a word $w\in W(m,n,k)$ and a partition $\gamma$. The
$q$-statistic on the word~$w$ again counts all ascents and descents of $w$ by
their positions. However, in Theorem~1, we see that the six possible
ascents/descents in $w$ are weighted differently by position:
$$
\align
&01 {\text{ by }}yz, \\
&02 {\text{ by }}z,\\
&10 {\text{ by }}x,\\
&12 {\text{ by }}xz,\\
&20 {\text{ by }}xy,\\
&21 {\text{ by }}y.\\
\endalign
$$
So if we choose $x=q^a$, $y=q^b$, $z=q^c$, an occurrence of $01$ in positions
$j$ and $j+1$ of $w$ contributes a weight of $q^{j(b+c)}$. This in turn implies
that the bijection $\phi$ must be modified so that the part in $\lambda$
corresponding to $w_j$ must be at least $b+c$ larger than the part in $\mu$
corresponding to $w_{j+1}$. We need six different inequalities for the six
possible juxtapositions of parts. Let $\phi_{a,b,c}$ be the modified bijection.

For example, if $m=k=2$, $n=1$, $a=2$, $b=c=1$, then the juxtaposed parts sizes
must differ by
$$
\align
&2 {\text{ for }}\lambda\mu,\\
&1 {\text{ for }}\lambda\theta,\\
&2 {\text{ for }}\mu\lambda,\\
&3 {\text{ for }}\mu\theta,\\
&3 {\text{ for }}\theta\lambda,\\
&1 {\text{ for }}\theta\mu.\\
\endalign
$$
The three possible triples $(\lambda,\mu,\theta)$ whose weight is $q^{12}$ are
given below, along with result of the bijection $\phi_{2,1,1}$:
$$
\align
&(22,6,11) \rightarrow (10022,31111),\\
&(32,5,11) \rightarrow (10022,22111),\\
&(43,1,22) \rightarrow (00221,21111).\\
\endalign
$$

\proclaim{Corollary 3} Let $a$, $b$ and $c$ be positive integers.
The generating function for all triples of partitions $(\lambda,\mu,\theta)$
without pairwise common parts, such that $\lambda$ has $m$ parts, $\mu$ has $n$
parts, and $\theta$ has $k$ parts, and any adjacent parts in the partition
$\lambda\cup\mu\cup\theta$ of type
\roster
\item $\lambda\mu$ differ by $b+c$,
\item $\lambda\theta$ differ by $c$,
\item $\mu\lambda$ differ by $a$,
\item $\mu\theta$ differ by $a+c$,
\item $\theta\lambda$ differ by $a+b$,
\item $\theta\mu$ differ by $b$,
\endroster
is given by
$$
\align
&\frac{q^{m+n+k}}{(q;q)_{m+n+k}}\!\biggl(q^{a(n+k)+bk}\!
\qtri{\!m+n+k-1\!}{m-1}{n}{k}{q^{a+b+c}}\!\!
{}+\\
&{}q^{b(m+k)+cm}\!\qtri{\!m+n+k-1\!}{m}{n-1}{k}{q^{a+b+c}}
+q^{c(n+m)+an}\!\qtri{\!m+n+k-1\!}{m}{n}{k-1}{q^{a+b+c}}\biggr).
\endalign
$$
\endproclaim

In Theorem 3, if all $x_i=q$, the following theorem results. All subscripts are
taken $\!\!\!\mod N$.

\proclaim{Theorem 4} The generating function for all $N$-tuples of integer
partitions
$(\lambda_1,\cdots, \lambda_N)$ without pairwise common parts, such that
\roster
\item"(a)" $\lambda_i$ has $a_i$ parts, $1\le i\le N$,
\item"(b)" if the partition $\lambda_1\cup\lambda_2\cup\cdots\cup\lambda_N$
has adjacent parts $bc$, for $b\in \lambda_i$ and $c\in \lambda_j$,
then $b-c\ge (i-j) \mod N$,
\endroster
is given by
$$
\frac{q^f}{(q;q)_f}\qbin{a_1+\cdots+a_N}{a_1,\cdots,a_N}{q^N}
\frac{\sum_{i=1}^N q^{e_i}}
{\sum_{i=0}^{N-1} q^{if}},
$$
where $f=a_1+a_2+\cdots+a_N$, and $e_i=a_i+2a_{i+1}+\cdots+(N-1)a_{i+N-2}$.
\endproclaim

{\subheading{5. Remarks}}

MacMahon \cite{5, \S30} defined a statistic related to $maj$, denoted here by
$MAJ$, which weights each descent by the amount of the descent. For example,
$$
MAJ(20211201)=2*1+1*3+2*6=17,
$$
because the descent $20$ in positions $1,6$ are weighted by $2-0=2$, while the
descent $21$ in position $3$ is weighted by $2-1=1$. Let $MIN$ denote the
analogous statistic using the ascents. Then MacMahon alludes \cite{5, \S40} to
the following theorem for words with three letters.

\proclaim{Theorem 5} For any non-negative integers $m$, $n$, and $k$ we have
$$
\align
\sum_{w\in W(m,n,k)} &x^{MAJ(w)}y^{MIN(w)}
=x^{n+2k}\qbin{m+n+k-1}{n}{xy}\qbin{m+k-1}{m-1}{(xy)^2}\\
&{}+y^{m-k}\qbin{m+n+k-1}{n-1}{xy}\qbin{m+k}{m}{(xy)^2}
\frac{(xy)^{2k}+(xy)^{m+k}}{1+(xy)^{m+k}}\\
&{}+y^{2m+n}\qbin{m+n+k-1}{n}{xy}\qbin{m+k-1}{m}{(xy)^2}.
\endalign
$$
\endproclaim

If $x=y$, $y=1$ or $x=1$, the three terms in Theorem~5 sum to a single product
(see \cite{5, \S38, \S40}). The proof of Theorem~5 is identical to the proof of
Theorem~1. We do not know a multivariable version of Theorem~5.


\Refs
\ref
\no1
\by G. Andrews
\book The Theory of Partitions
\publ Addison-Wesley
\publaddr Reading
\yr 1976
\endref
\ref
\no2
\by D. Foata and M.-P. Sch\"utzenberger
\paper Major index and inversion number of permutations
\jour Math. Nachr.
\vol 83
\yr 1978
\pages 143--158
\endref
\ref
\no3
\by J. Galovich and D. White
\paper Recursive statistics on words
\jour Disc. Math.
\vol 157
\yr 1996
\pages 169--191
\endref
\ref
\no4
\by P. MacMahon
\book Combinatory Analysis
\ed 3rd
\publ Chelsea
\publaddr New York
\yr 1984
\endref
\ref
\no5
\bysame
\paper The indices of permutations and the derivation therefrom of
functions of a single variable associated with the permutations of any
assemblage of objects
\ed 3rd
\jour Amer. J. Math.
\vol 35
\yr 1913
\pages 281--322
\endref
\endRefs
\enddocument
\end

