\documentclass[a5paper,landscape,fontsize=10pt]{scrartcl}
\usepackage{amsmath,amsthm,amsfonts,amssymb,verbatim}
\usepackage{graphicx}
\usepackage{pinlabel}
\usepackage[ngerman]{babel}  % German umlauts
\usepackage[utf8]{inputenc}  % German umlauts
\usepackage{hyperref}  % seems to be important for multicols
\pagenumbering{gobble}  % disable page numbering
\usepackage{multicol}  % split page in two parts

\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
%this is for my weird f

\usepackage{concmath}
%\usepackage{helvet} %NB this was the first font I tried

\renewcommand{\familydefault}{\sfdefault}
\usepackage[margin=0.2cm]{geometry}
\usepackage{parskip}

 \usepackage{picins}


\setlength\columnseprule{.4pt}


\usepackage{lipsum}
\let\OLDthebibliography\thebibliography
\renewcommand\thebibliography[1]{
  \OLDthebibliography{#1}
  \setlength{\parskip}{0pt}
  \setlength{\itemsep}{1pt plus 0.3ex}
}


\newtheorem*{namedtheorem}{\theoremname}
\newcommand{\theoremname}{testing}
\newenvironment{named}[1]{\renewcommand{\theoremname}{#1}
        \begin{namedtheorem}}
        {\end{namedtheorem}}

\usepackage{atbegshi}% http://ctan.org/pkg/atbegshi
\AtBeginDocument{\AtBeginShipoutNext{\AtBeginShipoutDiscard}}

% ! This is a major hack 
% cf https://tex.stackexchange.com/questions/140168/how-to-remove-a-blank-page-before-the-title-page
% I do not know how the first blank page was being generated; this throws it out
% Need to learn how to properly 'bleed' a photo

   \begingroup
\renewcommand{\section}[2]{}%
%\renewcommand{\chapter}[2]{}% for other classes
     
%%%%Extre stuff for H.T. (Vol 2 No 1)

\usepackage{tikz}
\usetikzlibrary{arrows.meta}

\tikzstyle{wall}=[line width=1pt]

\newcommand{\zvd}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (1.5,0)--(1.5,3);\protect\draw[wall] (0,2)--(1.5,2);\protect\draw[wall] (1.5,1)--(3,1);
\end{tikzpicture}
}

\newcommand{\zvu}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (1.5,0)--(1.5,3);\protect\draw[wall] (0,1)--(1.5,1);\protect\draw[wall] (1.5,2)--(3,2);
\end{tikzpicture}
}

\newcommand{\zhl}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (0,1.5)--(3,1.5);\protect\draw[wall] (2,0)--(2,1.5);\protect\draw[wall] (1,1.5)--(1,3);
\end{tikzpicture}
}

\newcommand{\zhr}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (0,1.5)--(3,1.5);\protect\draw[wall] (1,0)--(1,1.5);\protect\draw[wall] (2,1.5)--(2,3);
\end{tikzpicture}
}

\newcommand{\millr}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw (0,0) rectangle (4,4);\protect\draw[wall] (1,0)--(1,3);\protect\draw[wall] (0,3)--(3,3);\protect\draw[wall] (3,4)--(3,1);\protect\draw[wall] (4,1)--(1,1);
\end{tikzpicture}
}

\newcommand{\milll}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw (0,0) rectangle (4,4);
\protect\draw[wall] (3,0)--(3,3);\protect\draw[wall] (4,3)--(1,3);\protect\draw[wall] (1,4)--(1,1);\protect\draw[wall] (0,1)--(3,1);
\end{tikzpicture}
}


\newcommand{\pone}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (1.2,0)--(1.2,2.8);
\protect\draw[wall] (4,1.2)--(1.2,1.2);
\protect\draw[wall] (2.8,4)--(2.8,1.2);
\protect\draw[wall] (0,2.8)--(2.8,2.8);
\end{tikzpicture}
}

\newcommand{\ptwo}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (2.8,0)--(2.8,2.8);
\protect\draw[wall] (4,2.8)--(1.2,2.8);
\protect\draw[wall] (1.2,4)--(1.2,1.2);
\protect\draw[wall] (0,1.2)--(2.8,1.2);
\end{tikzpicture}
}

\newcommand{\pthr}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (2,0)--(2,4);
\protect\draw[wall] (0,1.2)--(2,1.2);
\protect\draw[wall] (2,2.8)--(4,2.8);
\end{tikzpicture}
}

\newcommand{\pfiv}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (2,0)--(2,4);
\protect\draw[wall] (0,2.8)--(2,2.8);
\protect\draw[wall] (2,1.2)--(4,1.2);
\end{tikzpicture}
}

\newcommand{\pfou}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (0,2)--(4,2);
\protect\draw[wall] (2.8,2)--(2.8,4);
\protect\draw[wall] (1.2,0)--(1.2,2);
\end{tikzpicture}
}

\newcommand{\psix}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (0,2)--(4,2);
\protect\draw[wall] (2.8,0)--(2.8,2);
\protect\draw[wall] (1.2,2)--(1.2,4);
\end{tikzpicture}
}

\newcommand{\psev}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (0,2)--(1.65,2);
\protect\draw[wall] (1.65,0)--(1.65,4);
%\protect\draw[wall] (1.2,0)--(2.8,0);
%\protect\draw[wall] (1.2,4)--(2.8,4);
\protect\draw[wall] (2.35,0)--(2.35,4);
\protect\draw[wall] (2.35,2)--(4,2);
\end{tikzpicture}
}

\newcommand{\peig}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.1275]
\protect\draw[wall] (2,0)--(2,1.65);
\protect\draw[wall] (0,1.65)--(4,1.65);
%\protect\draw[wall] (0,1.2)--(0,2.8);
%\protect\draw[wall] (4,1.2)--(4,2.8);
\protect\draw[wall] (0,2.35)--(4,2.35);
\protect\draw[wall] (2,2.35)--(2,4);
\end{tikzpicture}
}







\newcommand{\hvert}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[wall] (1,0)--(1,3);\protect\draw[wall] (2,0)--(2,3);\protect\draw[wall] (0,1.5)--(1,1.5);\protect\draw[wall] (2,1.5)--(3,1.5);
\end{tikzpicture}
}

\newcommand{\hhor}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[wall] (0,1)--(3,1);\protect\draw[wall] (0,2)--(3,2);\protect\draw[wall] (1.5,0)--(1.5,1);\protect\draw[wall] (1.5,2)--(1.5,3);
\end{tikzpicture}
}

\newcommand{\zsvd}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[fill=black!20] (1,0) rectangle (2,3);
\protect\draw[wall] (1,0)--(1,3);
\protect\draw[wall] (2,0)--(2,3);
\protect\draw[wall] (0,2)--(1,2);
\protect\draw[wall] (2,1)--(3,1);
\end{tikzpicture}
}

\newcommand{\zsvu}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[fill=black!20] (1,0) rectangle (2,3);
\protect\draw[wall] (1,0)--(1,3);
\protect\draw[wall] (2,0)--(2,3);
\protect\draw[wall] (0,1)--(1,1);
\protect\draw[wall] (2,2)--(3,2);
\end{tikzpicture}
}

\newcommand{\zshl}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[fill=black!20] (0,1) rectangle (3,2);
\protect\draw[wall] (0,2)--(3,2);
\protect\draw[wall] (0,1)--(3,1);
\protect\draw[wall] (1,2)--(1,3);
\protect\draw[wall] (2,0)--(2,1);
\end{tikzpicture}
}

\newcommand{\zshr}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[fill=black!20] (0,1) rectangle (3,2);
\protect\draw[wall] (0,2)--(3,2);
\protect\draw[wall] (0,1)--(3,1);
\protect\draw[wall] (2,2)--(2,3);
\protect\draw[wall] (1,0)--(1,1);
\end{tikzpicture}
}

\newcommand{\hv}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[wall] (1,0)--(1,3);
\protect\draw[wall] (2,0)--(2,3);
\protect\draw[wall] (1,1.5)--(2,1.5);
\end{tikzpicture}
}

\newcommand{\hh}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);
\protect\draw[wall] (0,1)--(3,1);
\protect\draw[wall] (0,2)--(3,2);
\protect\draw[wall] (1.5,1)--(1.5,2);
\end{tikzpicture}
}

\newcommand{\hor}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (0,1)--(3,1);\protect\draw[wall] (0,2)--(3,2);
\end{tikzpicture}
}

\newcommand{\ver}{
\begin{tikzpicture}[baseline=1.2mm,scale=0.17]
\protect\draw (0,0) rectangle (3,3);\protect\draw[wall] (1,0)--(1,3);\protect\draw[wall] (2,0)--(2,3);
\end{tikzpicture}
}




\def\P#1{\ifnum#1=1 $\pone$
\else\ifnum#1=2 $\ptwo$
\else\ifnum#1=3 $\pthr$
\else\ifnum#1=4 $\pfou$
\else\ifnum#1=5 $\pfiv$
\else\ifnum#1=6 $\psix$
\else\ifnum#1=7 $\psev$
\else\ifnum#1=8 $\peig$
 \else            \relax
\fi\fi\fi\fi\fi\fi\fi\fi}


%%%%%%%%%%%%%%

\begin{document}


%----------------
% 
\vspace*{-78mm}
\hspace*{-123mm}
\noindent
\includegraphics[scale=2.1]{postcard_image.pdf}
%
%----------------




\newpage

\vspace*{0mm}

\begin{multicols}{2}
    {\normalfont \bfseries Counting simple whirls}
    
 \medskip
    
    {\scriptsize Hugh Thomas} \\[-2pt]
    {\scriptsize Université du Québec à Montréal, Département de mathématiques}\\[-2pt]
    {\scriptsize \normalfont \ttfamily thomas.hugh\textunderscore r@uqam.ca}
    
    \medskip
    
{\normalfont \itshape November 1, 2024.} A rectangulation, like the one in the image, is a subdivision of a rectangle into subrectangles. There is a reasonable notion of what it means for two rectangulations to be combinatorially equivalent---{\bfseries say a word about what this is?}---and rectangulations are considered up to this equivalence.

Various patterns arise in a rectangulation, for example, there is an occurrence of a windmill $\ptwo$ in the image, centered at the blank square. A whirl is a rectangulation that avoids patterns of the form

\smallskip

$
\qquad\quad\P1\quad\P3\quad\P4\quad\P5\quad\P6\quad\P7\quad\P8
$

\smallskip

A whirl is peelable if it contains a rectangle that stretches from the left boundary to the right boundary, or from the top boundary to the bottom boundary. 
A simple whirl is a non-peelable whirl containing exactly one $\ptwo$ where the interior of the $\ptwo$ is not further divided. The rectagulation on the obverse provides an example of a simple whirl.
    
 Asinowski and Banderier \cite{AB} counted the simple whirls and showed that their generating function is $t^5C^4(t)$, where $C(t)$ is the well-known Catalan generating function\[C(t)=\frac{1-\sqrt{1-4t}}{2t}\]
   The Catalan generating function counts, in particular, planar binary trees. The image shows how a simple whirl can be decomposed into     
    
\columnbreak\par
    
 \parpic[r]{
 \framebox(60,75){\scriptsize %Postage ENLARGED THIS FOR A5
 }
 }
  four planar binary
  trees, each of whose roots has no left child---these four trees are shown beside the rectangulation, {\bfseries where two of the trees are single nodes (or childless roots).}
The nodes on the right correspond to the rectangles on the left. 
  
  This gives a bijective proof of the result of Asinowski and Banderier, and is part of a joint project-in-process with these authors.  


  
    
     
     \medskip
     
     \scriptsize
         \begin{thebibliography}{1}
           
\bibitem{AB}
Asinowski  \& Banderier.
\newblock {From geometry to generating functions: rectangulations and permutations}.
to appear.

\end{thebibliography}


\vfill
 \hfill {\scriptsize \, \textcopyright 2024 Mathematical Research Postcards}
    
\end{multicols} 


\end{document} 