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\author[C.~Banderier]{Cyril Banderier}
\author[C.~Krattenthaler]{Christian Krattenthaler}
\author[M.~Wallner]{Michael Wallner}
\title[Preface to the special issue for the 2021 Lattice Path Conference]{Preface to the special issue for \\ the 2021 Lattice Path Conference}

\begin{document}
\maketitle
\vspace{-10mm}
\section{The Lattice Path Conference}
\subsection{History of the conference}
Following an interdisciplinary interest for lattice paths after the publication in 1979 of the two books \textit{Lattice Path Combinatorics with Statistical Applications} by Tadepalli Venkata Narayana and \textit{Lattice Path Counting and Applications} by Sri Gopal Mohanty, Mohanty initiated a series of conferences on this topic.

\smallskip
\noindent
\begin{tabular}{ll}\setlength{\tabcolsep}{-2mm}
\renewcommand{\arraystretch}{1}
\hspace{.5mm}
	\begin{tabular}{c}\setlength{\tabcolsep}{-2mm}
		\includegraphics[width=4cm]{mohanty3}\\
		\small Sri Gopal Mohanty, father of\\ 
		\small the Lattice Path Conference.
	\end{tabular}
	\hspace{4.1mm}
	&\hspace{-3mm}
	\begin{minipage}{10.15cm}	
The first two events were held
at McMaster University in 1984 and 1990,
and then at the University of Delhi in 1994,
University of Vienna in 1998,
University of Athens in 2002,
East Tennessee State University in 2007, 
University of Siena in 2010, 
Cal Poly Pomona University in 2015,
Centre International de Rencontres Math\'ematiques in 2021.
	A new initiative started by dedicating the fourth conference in Wien to the memory of Germain Kreweras (1918--1998) and Tadepalli Venkata Narayana (1930--1987), both of whom made a significant contribution to the field. In the same spirit, the 2002 conference was dedicated to the memory of Istv\'an Vincze (1912--1999). The 2015 conference was dedicated to Shreeram Shankar Abhyankar (1930--2013), Philippe Flajolet (1948--2011), and Lajos Tak\'acs (1924--2015).
	\end{minipage}\hspace{-3mm}
\end{tabular}

\smallskip
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For each conference there is the tradition to have a journal special issue where anyone could submit work related to lattice paths.\footnote{See the website 
of the \href{https://lipn.fr/~banderier/LPC/2021/}{ Lattice Path Conference} for links to these special issues and further information.}
We shall come back to the current special issue in Section~\ref{sec:specialissue}.
Before, let us say a few words about the event itself.


\subsection{The 2021 Lattice Path Conference}

The $9$th Lattice Path Conference was held in hybrid form (given the circumstances of the global Covid pandemic) during 21--25 June 2021 at the Centre International de Rencontres Math\'ematiques (CIRM) in Luminy, France. This event attracted 227 registered participants with affiliations from the ``four corners of the world'' from 34 different countries
of which 24 attended in person. 
Moreover, about 40 more people attended casually online without being registered (we made the event fully open), 
and some colleagues even organized some watch parties in their university, so the real number of participants in the event will remain unknown.
%(actually, from 34 different countries: Australia, Austria, Canada, China, Columbia, Czech Republic, Finland, France, Germany, Greece, Hungary, India, Iceland, Ireland, Israel, Italy, Japan, Madagascar, Mexico, Netherlands, New Zealand, Poland, Portugal, Russia, Saudi Arabia, South Africa, South Korea, Spain, Sweden, Switzerland, Taiwan, Tunisia, UK, USA),

\noindent
\begin{tabular}{@{}c}
	\newcommand{\picstripheight}{3.3191cm}%
	\includegraphics[height=\picstripheight]{hybrid1}%
	\includegraphics[height=\picstripheight]{hybrid2}%
	\includegraphics[height=\picstripheight]{hybrid3}%
\\
{\small As visible via some details in the above photos,}\\
{\small the 2021 Lattice Path Conference was a hybrid conference.}
\end{tabular}
\bigskip

The conference consisted of $27$ invited talks and $22$ accepted posters, leaving also time for scientific discussions.
The presentations covered a wide range of topics, including 
\begin{itemize}
 \item algebraic combinatorics (Young tableaux, representation theory, symmetric polynomials, representation theory, \dots),
 \item analytic combinatorics (kernel method, symbolic method, asymptotics, \dots),
 \item enumerative combinatorics (walks in convex and non-convex cones, alternating sign matrices, generating functions, bijections, \dots),
 \item probability theory (Brownian motion, Markov chains, random trees, \dots),
 \item computer algebra (orthogonal polynomials, holonomy theory, \dots),
		\item theoretical computer science (random generations, context-free grammars, \dots),
 \item number theory (integer partitions, continued fractions, \dots),
 \item theoretical physics (Liouville quantum gravity, square ice, \dots).
\end{itemize}

The conference was a great success, with many stimulating talks and discussions. 
We would like to thank all of the participants, including the invited speakers and the attendees, for their contributions to the conference. 
We would also like to extend our gratitude to the staff at CIRM, who, like always, was warmly welcoming,
even in this difficult period: CIRM reopened for our meeting after many months of closure due to the Covid containment and the interdiction of meetings.
Finally, we would like to thank Kilian Raschel who was the main sponsor of this event
via his ERC starting grant COMBINEPIC, which allowed us to help CIRM who financially suffered a lot of so many months of closure.
 %no other sponsors : other instutions just paid the trip / hotel of 2-3 members of their labs, they did not sponsor "other people".


This event successfully brought together leading experts in the field and contributed to the development of new
research directions. Several conjectures were even solved during the event! 
We hope that our special issue dedicated to this conference will serve as a valuable
resource for researchers in combinatorics, probability theory, and statistical physics, and
that they will inspire further research on lattice paths.

Let us now give the lists of organizers, speakers, and participants of this meeting.

\subsection{Scientific and organizing committee}
\begin{itemize}
	\item Cyril Banderier (CNRS, Universit\'e Sorbonne Paris Nord)
	\item Jehanne Dousse (CNRS, Universit\'e Lyon 1)
	\item Enrica Duchi (Universit\'e Paris Diderot)
	\item Christian Krattenthaler (Universit\"at Wien)
	\item Greta Panova (University of Southern California)
	\item Kilian Raschel (CNRS, Universit\'e de Tours)
	\item Michael Wallner (TU Wien)
\end{itemize}
\pagebreak
\subsection{Invited talks}~\\[4mm]
\label{sec:talks}
\def\myheight{23.1mm}
\hspace{-5mm}
\begin{tabular}{c@{}l}
\begin{tabular}{c}\ \ \includegraphics[height=\myheight]{George_Andrews} \ \ \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} George Andrews (Pennsylvania State University):\\ \emph{Schmidt type partitions and partition analysis } \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Andrei_Asinowski} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Andrei Asinowski (Alpen-Adria-Universit\"at Klagenfurt):\\ \emph{Vectorial kernel method and lattice paths with patterns} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Philippe_Biane} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Philippe Biane (CNRS, Universit\'e Paris-Est):\\ \emph{Mating of discrete trees and walks in the quarter-plane} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Alin_Bostan} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm}Alin Bostan (INRIA, Saclay):\\ \emph{How to prove or disprove the algebraicity of a generating function using a computer} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Mireille_Bousquet-Melou} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Mireille Bousquet-M\'elou (CNRS, Universit\'e de Bordeaux):\\ \emph{Invariants for walks avoiding a quadrant} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Timothy_Budd} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Timothy Budd (Radboud University):\\ \emph{Winding of simple walks on the square lattice} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Philippe_di_Francesco} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Philippe Di Francesco (University of Illinois at Urbana-Champaign and IPhT Saclay):\\ \emph{Triangular ice: combinatorics and limit shapes} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Sergi_Elizalde} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Sergi Elizalde (Dartmouth College):\\ \emph{Counting lattice paths by the number of crossings and major index} \end{minipage}\end{tabular}\\
\begin{tabular}{c}\includegraphics[height=\myheight]{Ilse_Fischer} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Ilse Fischer (University of Vienna):\\ \emph{The alternating sign matrices/descending plane partitions relation:\\ n+3 pairs of equivalent statistics} \end{minipage}\end{tabular}\\
\end{tabular}
\clearpage


\hspace{-5mm}
\def\myheight{23.9mm}
\begin{tabular}{ll}
\begin{tabular}{l}\includegraphics[height=\myheight]{Ira_Gessel} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Ira Gessel (Brandeis University):\\ \emph{Redundant generating functions in lattice path enumeration} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Vadim_Gorin} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm}Vadim Gorin (MIT):\\ \emph{Addition of matrices at high temperature} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Tony_Guttmann} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Tony Guttmann (Melbourne University):\\ \emph{Extracting asymptotics from series coefficients } \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Nina_Holden} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Nina Holden (ETH Z\"urich):\\ \emph{Random triangulations and bijective paths to Liouville quantum gravity} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Mourad_Ismail} \end{tabular} & \begin{tabular}{l@{}} \begin{minipage}{12.5cm}Mourad E.H. Ismail (University of Central Florida):\\ \emph{Orthogonal polynomials, moments, and continued fractions} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Satya_Majumdar} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Satya Majumdar (CNRS, Universit\'e Paris Sud):\\ \emph{Nonintersecting Brownian bridges in the flat-to-flat geometry} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Olya_Mandelshtam} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Olya Mandelshtam (University of Waterloo):\\ \emph{A Markov chain on tableaux that projects to a multispecies totally asymmetric zero range process} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Irene_Marcovici} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Ir\`ene Marcovici (Universit\'e de Lorraine):\\ \emph{Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Stephen_Melczer} \end{tabular} & \begin{tabular}{@{}l} \begin{minipage}{12.5cm} Stephen Melczer (University of Pennsylvania):\\ \emph{Lattice walks and analytic combinatorics in several variables} \end{minipage}\end{tabular}\\
\end{tabular}
\clearpage

\def\myheight{23.1mm}
\hspace{-5mm}
\begin{tabular}{ll}
\begin{tabular}{l}\includegraphics[height=\myheight]{Robin_Pemantle} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Robin Pemantle (University of Pennsylvania):\\ \emph{Generating function technologies: applications to lattice paths} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Bruno_Salvy} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Bruno Salvy (INRIA / ENS Lyon):\\ \emph{Computation of tight enclosures for Laplacian eigenvalues} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Michael_Singer} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Michael Singer (North Carolina State University):\\ \emph{Differentially algebraic generating series for walks in the quarter plane} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Perla_Sousi} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Perla Sousi (University of Cambridge):\\ \emph{The uniform spanning tree in 4 dimensions} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Andrea_Sportiello} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Andrea Sportiello (CNRS, Universit\'e Paris Nord):\\ \emph{Boltzmann sampling in linear time: irreducible context-free structures} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Xavier_Viennot} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Xavier Viennot (CNRS, Universit\'e de Bordeaux):\\ \emph{Heaps and lattice paths} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Karen_Yeats} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Karen Yeats (University of Waterloo):\\ \emph{\L ukasiewicz walks and generalized tandem walks} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Doron_Zeilberger} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Doron Zeilberger (Rutgers University):\\ \emph{Using symbolic dynamical programming in lattice paths combinatorics} \end{minipage}\end{tabular}\\
\begin{tabular}{l}\includegraphics[height=\myheight]{Paul_Zinn-Justin} \end{tabular} & \begin{tabular}{l} \begin{minipage}{12.5cm} Paul Zinn-Justin (Melbourne University):\\ \emph{Generalized pipe dreams and lower-upper scheme} \end{minipage}\end{tabular}\\
\end{tabular}

\smallskip
\noindent \rule{3cm}{0.3pt}\\
\noindent A recording of all the talks can be found at \url{https://lipn.fr/~banderier/LPC/2021/}.

\clearpage
\subsection{Accepted posters}
\begin{itemize}[leftmargin=2.52mm,itemsep=3pt]
 \item Ault Shaun, Charles Kicey: \emph{From lattice paths to standard Young tableaux}
 \item Cyril Banderier, Marie-Louise Lackner, Michael Wallner: \emph{Latticepathology and symmetric functions}
 \item Nicholas Beaton: \emph{Walks obeying two-step rules on the square lattice}
 \item Swee Hong Chan, Igor Pak, Greta Panova: \emph{Log-concavity in posets and random walks}
 \item Sergey Dovgal, Mohamed Lamine Lamali, Philippe Duchon: \emph{A phase transition in non-deterministic walks with two or more variables}
 \item Andrew Elvey Price: \emph{Enumeration of walks with small steps by winding angle}
 \item Rigoberto Fl{\'o}rez, Jos\'e L.~Ram{\'\i}rez, Fabio A. Velandia, Diego Villamiza: \emph{Restricted Dyck paths}
 \item Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, Alan Sokal: \emph{Coefficientwise total positivity of some matrices defined by linear recurrences}
 \item Hans H\"ongesberg: \emph{Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids}
 \item Heba Ayeda, David Beecher, Alan Krinik, Jeremy J. Lin, David Perez, Thuy Vu Dieu Lu, Weizhong Wong: \emph{Lattice paths with alternating probabilities}
 \item Josef K\"ustner, Michael Schlosser, Meesue Yoo: \emph{Lattice paths and negatively indexed weight-dependent binomial coefficients}
 \item Florian Lehner, Christian Lindorfer, Wolfgang Woess: \emph{The language of self-avoiding~walks}
 \item Satya Majumdar, Francesco Mori, Gregory Schehr: \emph{Distribution of the time between maximum and minimum of random walks}
 \item St\'ephane Ouvry, Alexios Polychronakos, Shuang Wu: \emph{Algebraic area counting for lattice closed random walks}
 \item Alan Krinik, Gerardo Rubino: \emph{The exponential-dual matrix method: applications to Markov chain analysis}
 \item Andrei Asinowski, Benjamin Hackl, Sarah Selkirk: \emph{Down-step statistics in generalized Dyck paths}
 \item Myrto~Kallipoliti,~Robin~Sulzgruber,~Elini~Tzanaki:~\emph{Patterns~in~Shi~tableaux~and~Dyck~paths}
 \item Malvina Vamvakari: \emph{On $q$-order statistics}
 \item Florian Aigner, Gabriel Frieden: \emph{qRSt: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials}
 \item Quang-Nhat Le, Sinai Robins, Christophe Vignat, Tanay Wakhare: \emph{A continuous analogue of lattice path enumeration}
 \item Jisun Huh, Sun-Young Nam, Meesue Yoo: \emph{LLT polynomials in a nutshell: on Schur expansion of LLT polynomials}
 \item Benjamin De Bruyne, Satya Majumdar, Gregory Schehr: \emph{Generating discrete-time constrained random walks}.
%\href{https://arxiv.org/pdf/2104.06145.pdf}{\emph{Generating discrete-time constrained random walks}}
\end{itemize}
The poster session was organized on the online platform \href{https://app.gather.town/app/ESXz27O6PQeCeaVG/LatticePathConference}{Gather.town},
where all the posters can still be perused.

\clearpage

\subsection{The Hotel Latticepathologia escape game}
The image below shows a part of the ``Hotel Latticepathologia'' 
(a virtual location designed by Cyril Banderier, Jehanne Dousse, and Michael Wallner),
where the poster session was taking place.

\noindent
	\includegraphics[width=0.495\textwidth,trim={0 12mm 0 0},clip]{poster1}
	\includegraphics[width=0.495\textwidth,trim={0 12mm 0 0},clip]{poster2}

This virtual place gave the online participants the opportunity to socialize
and to participate in an escape game! 
In its first part, you have to identify a mysterious mathematician in each room in the 
Hotel Latticepathologia ({\small{\url{https://tinyurl.com/s6njbsku}}}),
using clues scattered throughout. Will you succeed? 

In its second part, you have to solve
a collection of puzzles designed in collaboration with Vivien Ripoll; see \url{https://lipn.fr/~cb/LPC/2021/Puzzles/}.
It includes a musical concert by 9 conference participants. Don't miss it!
 
\smallskip
Below is a sample puzzle from the escape game. Can you solve it? 
\bigskip

\noindent
\includegraphics[width=\textwidth,trim={15mm 16mm 10mm 15mm},clip]{2_ConfusingDream}%
\clearpage

\subsection{Registered participants}
{\small 
Most of the 227 registered participants were online.
A few of them had the opportunity to enjoy CIRM's beautiful surroundings.
}

\vspace{1mm}
\noindent
\begin{center}
\begin{tabular}{c}
	\newcommand{\picstripheight}{2.6cm}%
	\includegraphics[height=\picstripheight]{cirm1}%
	\quad
	\includegraphics[height=\picstripheight]{cirm2}%
\end{tabular}
\end{center}

\vspace{1mm}

{
\footnotesize
List of registered participants:
David Adame-Carrillo, Mohammed Ageel, Florian Aigner, Marie Albenque, Seamus Albion, 
Ian Alevy, Irha Ali, George Andrews, Omer Angel, Margaret Archibald, David Ash, 
Andrei Asinowski, Shaun Ault, Jean-Christophe Aval, Heba Ayeda, Arvind Ayyer, 
Be\'ata B\'enyi, Cyril Banderier, Josaphat Baolahy, Elena Barcucci, Jean-Luc Baril, 
Erik Bates, Nicholas Beaton, David Beecher, Chiheb Ben Bechir, Sudip Bera, 
Olivier Bernardi, Antonio Bernini, David Bevan, Philippe Biane, Arthur Blanc-Renaudie, 
Aubrey Blecher, Alin Bostan, Mireille Bousquet-M\'elou, C\'edric Boutillier, J\'er\'emie Bouttier, 
Timothy Budd, Th\'eophile Buffi\`ere, Ariane Carrance, Giulio Cerbai, Swee Hong Chan, Linxiao Chen, 
Shaoshi Chen, Fr\'ed\'eric Chyzak, Lapo Cioni, Alice Contat, Michael Coopman, Sylvie Corteel, Logan Crew, 
Cesar Cuenca, Nicolas Curien, St\'ephane Dartois, Benjamin De Bruyne, Bishal Deb, Nachum Dershowitz, 
Hiranya Kishore Dey, Philippe Di Francesco, Lucia Di Vizio, Ruiwen Dong, Robert Donley, 
Jehanne Dousse, Sergey Dovgal, Thomas Dreyfus, Michael Drmota, Enrica Duchi, Dennis Eichhorn, 
Sergi Elizalde, Andrew Elvey Price, Sen-Peng Eu, Wenjie Fang, Valentin F\'eray, Luca Ferrari, 
Ilse Fischer, Rigoberto Fl{\'o}rez, Luis Fredes, Gabriel Frieden, \'Eric Fusy, Ira Gessel, 
Sudhir Ghorpade, Juan Gil, Tomack Gilmore, Vadim Gorin, Adam Gregory, Tony Guttmann, Hans H\"ongesberg, 
Benjamin Hackl, Aliakbar Haghighi, Eva-Maria Hainzl, Charlotte Hardouin, Kilian Hermann, 
Clemens Heuberger, Pawel Hitczenko, Hung Hoang, Nina Holden, Sam Hopkins, Yueyun Hu, Justin Hua, 
Mourad~E.~H.~Ismail, Svante Janson, Helen Jenne, Fr\'ed\'eric Jouhet, Josef K\"ustner, Wonwoo Kang, 
Manuel Kauers, Rinat Kedem, Ghizlane Kettani, Mikhail Khristoforov, Charles Kicey, 
Donghyun Kim, Sergey Kirgizov, Victor Kleptsyn, Arnold Knopfmacher, Isaac Konan, 
Irina Kourkova, Christian Krattenthaler, Alan Krinik, Nishu Kumari, Raunak Kundagrami, 
Florian Lehner, Helder Lima, Zhicong Lin, Christian Lindorfer, Martin Loebl, Baptiste Louf, 
Torsten M\"utze, Satya Majumdar, Pritam Majumder, Olya Mandelshtam, Jean-Francois Marckert, 
Ir\`ene Marcovici, Barbara Margolius, Hana Mel\'anov\'a, Stephen Melczer, Laurent Menard, 
Sri Gopal Mohanty, Derrick Mohlala, Francesco Mori, Lukas Nabergall, Philippe Nadeau, 
Victor Nador, Mehdi Naima, Hiroshi Naruse, Andreas Nessmann, David Nguyen, 
Hadrien Notarantonio, Soichi Okada, St\'ephane Ouvry, J.~E.~Paguyo, Nimisha Pahuja, 
Greta Panova, Jay Pantone, Eveliina Peltola, Robin Pemantle, Karol Penson, Leonid Petrov, 
Renzo Pinzani, Thomas Prellberg, Helmut Prodinger, Sanjay Ramassamy, Jos\'e Ram{\'\i}rez, 
Kilian Raschel, Vivien Ripoll, Tom Roby, Martin Rubey, Gerardo Rubino, Tiadora Ruza, 
Nasser Saad, Bruce Sagan, Manjil Pratim Saikia, Bruno Salvy, Yoshio Sano, Gilles Schaeffer, 
Gregory Schehr, Michael Schlosser, Jeanne Scott, Blair Seidler, Sarah Selkirk, 
Timo Seppalainen, Michael F.~Singer, Alexandros Singh, Erik Slivken, Rebecca Smith, 
Alan Sokal, U-Keun Song, Perla Sousi, Andrea Sportiello, Richard Stanley, Dennis Stanton, 
Benedikt Stufler, Adrian Tanasa, Benjamin Terlat, Vasu Tewari, Paul Th\'evenin, 
Mikhail Tikhonov, Jordan Tirrell, Jessica Tomasko, Joonas Turunen, Eleni Tzanaki, 
Malvina Vamvakari, Roger Van Peski, Zo\'e Varin, Ekaterina Vassilieva, Fabio Velandia, 
Xavier Viennot, Christophe Vignat, Diego Villamizar, Michael Voit, Trung Vu, David Wahiche, 
Tanay Wakhare, Michael Wallner, Harriet Walsh, Guoliang Wang, Sebastian Wild, 
Mark Wilson, Peter Winkler, Wolfgang Woess, Elaine Wong, Zaidan Wu, Karen Yeats, 
Meesue Yoo, Sergey Yurkevich, Doron Zeilberger, Noam Zeilberger, Jiang Zeng, Yan Zhuang, Paul Zinn-Justin.
}

\clearpage

\section{Our special issue in the ``S\'eminaire Lotharingien de Combinatoire''}
\label{sec:specialissue}

The ``S\'eminaire Lotharingien de Combinatoire'' is an international biannual 
seminar, cofounded in 1980 by Dominique Foata (Strasbourg), Adalbert Kerber (Aachen and Bayreuth), and Volker Strehl (Erlangen). 
The name of the seminar comes from the fact that these cities were almost covered by Lotharingia, a part of the Carolingian Empire.
In~1994, an eponymous journal was launched %founded 
and quickly gathered a wider international audience,
welcoming also articles independent of any participation to the actual seminar.
On some occasions, the journal has had special issues dedicated to Festschriften or conference proceedings.
All volumes are freely accessible at~\url{https://www.mat.univie.ac.at/~slc/}.

We are pleased to add another volume
to this collection of the journal of the S\'eminaire Lotharingien de Combinatoire.
Our volume, dedicated to the themes of the 2021 Lattice Path Conference, contains the following contributions:

{
\newcommand{\paper}[2]{\item #1: \emph{#2}}
\begin{enumerate}[label={\arabic*}.]
\setlength\itemsep{1mm}
	\paper{Helmut Prodinger}{A walk in my lattice path garden}

	\paper{Anthony J.\ Guttmann and V\'aclav Kot\v{e}\v{s}ovec}{A numerical study of $L$-convex polyominoes and 201-avoiding ascent sequences}
	
	\paper{David W.\ Ash}{Introducing DASEP: the doubly asymmetric simple exclusion process}
	
	\paper{St\'ephane Ouvry and Alexios P.\ Polychronakos}{Signed area enumeration for lattice paths}
	
	\paper{Aubrey Blecher and Arnold Knopfmacher}{Left-to-right maxima in Dyck paths}
	
	\paper{Rigoberto Fl\'orez, Toufik Mansour, Jos{\'e} L.~Ram\'\i rez, Fabio A.~Velandia, and Diego Villamizar}{Restricted Dyck paths on valleys sequence}
	
	\paper{Sergi Elizalde}{Counting lattice paths by crossings and major index II: tracking descents via two-rowed arrays}
	
	\paper{Jang Soo Kim and Dennis Stanton}{Three families of $q$-Lommel polynomials}
	
	\paper{Malvina Vamvakari}{On $q$-order statistics}
	
	\paper{Thomas Dreyfus}{Differential algebraic generating series of weighted walks in the quarter plane}
	
	\paper{Rodolphe Garbit and Kilian Raschel}{The generating function of the survival probabilities in a cone is not rational}
		
	\paper{Rafik Aguech, Asma Althagafi, and Cyril Banderier}{Height of walks with resets, the Moran model, and the discrete Gumbel distribution}

%	\paper{Axel Bacher}{Generalized Dyck paths of bounded height }

%	\paper{Andrea Sportiello} {Boltzmann sampling of irreducible context-free structures in linear time.}
\end{enumerate}
}

We would like to thank all the authors for their patience during the interactions with the editors,
which (hopefully!) resulted in a pleasantly polished volume\dots\  
Last but not least, we would like to also thank all the referees for their excellent work. 
We hope that the reader will enjoy this volume and will eventually be motivated to contribute to the next Lattice Path Conference! 
\clearpage

\section{A panorama on lattice paths}

Let us end this preface with two beautiful photos taken during a random walk in the proximity of the conference location
(warm thanks to Andreas Nessmann and Sergey Dovgal for sharing them with us).

\begin{center}
\begin{tabular}{c}
	\includegraphics[width=13cm]{path0}
	\\
	\includegraphics[width=13cm]{path2}
\end{tabular}
\end{center}
\bigskip
\begin{center}
A path\dots\ what else could end our preface?
\end{center}
\end{document}