%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[submission]{FPSAC2026}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}
\usepackage{mathtools}  % даёт dcases, dcases*
\usepackage{amssymb}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem*{lemma*}{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% new word
\newcommand{\new}[1]{\textbf{\textit{#1}}}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{calc}

% bold
\newcommand\vi{\bf{i}}
\newcommand\istd{\bf{i}_{\text{std}}}
\newcommand\bh{{\bf h}}
\newcommand\bnu{\bm{\nu}}

% chalk
\newcommand\Cq{\mathbb C(q)}
\newcommand\C{\mathbb C}
\newcommand\CC{\mathbb C}
\newcommand\Qq{\mathbb Q(q)}
\newcommand\Q{\mathbb Q}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

% cal
\newcommand\cA{\mathcal A}
\newcommand{\T}{\mathcal{T}}
\newcommand\cT{\mathcal{T}} % Eventually replace \T by \cT 
\newcommand{\K}{\mathcal{K}} 
\newcommand\cK{\mathcal{K}} % ditto 
\newcommand\cI{\mathcal{I}}
\newcommand{\cL}{\mathcal{L}}


% frak
\newcommand{\p}{\mathfrak{p}}
\newcommand{\Sn}{\mathfrak{S}}
\newcommand\gl{\mathfrak{gl}}

\newcommand{\varnu}{\nu}  

\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\KPF}{KPF}
\DeclareMathOperator{\ch}{char}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\dpro}{dpro}

\def\dash#1{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle#1$}%
  \stackengine{1.2\LMpt}{$\SavedStyle#1$}{\rule{\wd0}{1\LMpt}}{O}{c}{F}{F}{S}%
}}

\usepackage{lipsum}

%% define your title in the usual way
\title[Asymptotics of generalized Tesler matrices]{Asymptotics of generalized Tesler matrices \footnote{This paper is an extended abstract of I.~Balashov, C.~Bulavenko, and Y.~Molybog, ``Tesler matrices and Lusztig data,'' \emph{Electron. J. Combin.} \textbf{33} (2026), no.~2, Paper No.~P2.25, \href{https://doi.org/10.37236/13427}{doi:10.37236/13427}.} }

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[I. Balashov, C. Bulavenko, and Y. Molybog]{Ivan Balashov, Constantine Bulavenko, and Yaroslav Molybog}

%% 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}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{This paper studies the asymptotic enumeration of generalized Tesler matrices. To suit this goal, we introduce integral Tesler matrices, linking Tesler matrices to integer partitions and obtaining sharp logarithmic asymptotics for flat hook sum vectors. 
We then introduce height diagrams to decompose more complicated hook sum vectors into flat pieces. Thereby, we establish the asymptotics of Tesler matrices with staircase hook sum vectors, significantly improving the best known bounds. In particular, the logarithmic asymptotic for the number of regular Tesler matrices is obtained. }

%% put your French abstract here, or comment this out if you don't have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Tesler matrices, Kostant partition function, asymptotic enumeration}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\articlenumber{38}
\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

% \section{A section}

% \lipsum[3]

% \begin{equation}\label{eqn:eq1}
% \int_0^\infty \exp(-x) \, \mathrm{d}x = 1.
% \end{equation}

% \begin{lem}\label{lem:lem1}
% \lipsum[11]
% \end{lem}

% \lipsum[10]

% \begin{figure}
% \centering
% \includegraphics[width=0.5\textwidth]{plot}
% \caption{A plot of a function.}
% \label{fig:plot}
% \end{figure}

% \section{A subsection}

% \lipsum[5]

% \begin{align*}
% 1 + 1 + 1 &= 2 + 1 \\ &= 3.
% \end{align*}

% \subsubsection{A subsubsection}

% \lipsum[17]

% Here is a citation~\cite{greenwade93} with URL. Here is another
% citation~\cite{MR4}, which is the earliest-numbered Math Review with a
% functioning DOI: MR0000004. Finally, a reference to
% equation~\eqref{eqn:eq1}. And a reference to Figure~\ref{fig:plot}.

\section*{Introduction}

Generalized Tesler matrices are upper-triangular matrices with non-negative integer entries that arise in the study of algebraic combinatorics, polyhedral geometry, and representation theory. The number of those is also known in Lie theory as the values of the type $A_n$ Kostant partition function ($\KPF$). Still, even for the number of regular Tesler matrices $T\left(\mathbf{1}^n\right)$, the asymptotic estimates were far from sharp. In this paper we drastically improve the previous benchmark from \cite{morales} by combining our results for Tesler matrices of flat and staircase hook sums.

In \Cref{MMM}, we introduce \new{integral Tesler matrices} and obtain asymptotics for flat hook sum vectors [\Cref{Theo}, \Cref{theo2}, \Cref{theo3}]. Then, in \Cref{ovs_pid}, we utilize the Kostant picture representation and introduce \new{height diagrams}, which we use to \new{decompose} general hook sum vectors (weights for $\KPF$) into flat pieces. In particular, we obtain asymptotics for Tesler matrices with `staircase' hook sums $\bh=(h_1,\dots,h_n),\,h_k>0$ (\Cref{CorSum}). This, notably, leads to the obtaining (\Cref{ATesler}) of $$\ln T(\mathbf{1}^n)\sim\sqrt{\frac{8}{27}}\,\pi n\sqrt{n}.$$ 

% \section{Introduction}

% We are concerned with the enumeration of generalized Tesler matrices and, equivalently, with the type $A_n$ Kostant partition function because of their significant role in algebraic combinatorics, polyhedral geometry, and representation theory.

% First, we introduce \new{integral Tesler matrices}, a new representation that links Tesler matrices to integer partitions and, in particular, enables us to obtain asymptotics for `flat' hook sum vectors. 
% The combined result of [\Cref{Theo}, \Cref{theo2}, \Cref{theo3}] is
%     $$
%         \ln T\underbrace{\left(H,\,0,\,0,\,\dots,\,0\right)}_{W\text{ times}}\sim\begin{cases}
%             W^2 \ln \left(\frac{\sqrt H} W \right) & \text{ if } W \prec \sqrt H \\
%             \Theta\left(H\right) & \text{ if } W \asymp \sqrt H \\
%             \sqrt{\frac{2}{3}}\pi W \sqrt H & \text{ if } W \succ \sqrt H 
%         \end{cases}
%     $$

% Then we exploit the Kostant pictures representation to introduce \new{height diagrams}, a tool used to \new{decompose} general hook sum vectors (weights for $\KPF$) into the aforementioned `flat' ones. Notably, we derive the asymptotics for Tesler matrices with `staircase' hook sum vectors ($\bh=(h_1,\dots,h_n),\,h_k>0$) in \Cref{CorSum}. In particular, we prove $\ln T(\mathbf{1}^n)\sim\sqrt{\frac{8}{27}}\,\pi n\sqrt{n}$ (\Cref{ATesler}). Taken together, our results for flat and regular hook sum vectors drastically improve the previous benchmark from \cite{morales}.

\section{Generalized Tesler matrices and their asymptotics} \label{MMM}
\subsection{Generalized Tesler matrices and integral Tesler matrices}
\label{ss:T+I}
First of all, we shall define Tesler matrices. Let $U_n$ be the set of $n\times n$ upper-triangular matrices with non-negative integer entries.
% 
\begin{definition}
    For a matrix $A = \left(a_{ij}\right) \in U_n$ we define its $k$\textsuperscript{th} \new{hook sum} $h_k$ (\Cref{pcthooksums}) as
    $$
        h_k=\sum_{i=k}^n{a_{ki}}-\sum_{i=1}^{k-1}{a_{ik}},\,1\leq k\leq n,
    $$ 
    and its \new{hook sum vector} as $\mathbf{ h}=\left(h_1,\,h_2,\,\ldots,\,h_n\right)$.
\end{definition}
\begin{figure}[ht]
    \centering
    \small
\begin{equation*}
\begin{bmatrix}
\color{red} 0 & \color{red} 0 & \color{red} 1 & \color{red} 0 & \color{red} 0\\
 & 1 & 0 & 0 & 0\\
 &  & 0 & 1 & 1\\
&  &  & 0 & 2\\
&  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 & \color{blue} 0 & 1 & 0 & 0\\
 & \color{red} 1 & \color{red} 0 & \color{red} 0 & \color{red} 0\\
 &  & 0 & 1 & 1\\
 &  &  & 0 & 2\\
 &  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 & 0 & \color{blue} 1 & 0 & 0\\
 & 1 & \color{blue} 0 & 0 & 0\\
 &  & \color{red} 0 & \color{red} 1 & \color{red} 1\\
 &  &  & 0 & 2\\
 &  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 1 & \color{blue} 0 & 0\\
 & 1 & 0 & \color{blue} 0 & 0\\
 &  & 0 & \color{blue} 1 & 1\\
 & &  & \color{red} 0 & \color{red} 2\\
 &  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 & 0 & 1 & 0 & \color{blue} 0\\
 & 1 & 0 & 0 & \color{blue} 0\\
 & & 0 & 1 & \color{blue} 1\\
 & & & 0 & \color{blue} 2\\
 &  &  &  & \color{red} 4
\end{bmatrix}
\end{equation*}
\caption{We add the \textcolor{red}{red} entries and subtract the \textcolor{blue}{blue} ones to get $\mathbf{h}=\left(1,\,1,\,1,\,1,\,1\right)$}
    \label{pcthooksums}
\end{figure}
Then the set of all $U_n$ matrices with a given hook sum vector $\mathbf{h}$ is called the set of \new{(generalized) Tesler matrices with hook sum $\mathbf h$} and is denoted as $\T\left(\mathbf{h}\right)$, while $T\left(\mathbf{h}\right)=\left|\T\left(\mathbf{h}\right)\right|$.
% 
\begin{example}\label{Tex}
    % If we have the hook sum to be 
    When the hook sum is $\left(1,\,1,\,\ldots,\,1\right)=\mathbf{1}^n$, then $\T\left(\mathbf{1}^n\right)$ is called the set of \new{regular Tesler matrices}. For $n=2$ we have only two regular Tesler matrices:
\begin{align*}
\begin{bmatrix}
1 & 0 \\
& 1 
\end{bmatrix}\;\;\textsc{and}\;\;
\begin{bmatrix}
0 & 1 \\
& 2 
\end{bmatrix}.
\end{align*}
\end{example}

We are concerned with counting the number of (regular) Tesler Matrices. However, the definition above uses subtraction, which makes it awkward in combinatorial proceedings. It is natural that we address this flaw before we proceed with further analysis.

Consider the map $I:U_n\to U_n$ that sends $\left[a_{ij}\right]$ into $\left[b_{ij}\right]$ such that $b_{ij}=a_{i\left(j+1\right)},\,j\neq n$ and $b_{in}=a_{ii}$. Then $I\left(\T\left(\mathbf{h}\right)\right)$ has an alternative definition as follows.

\begin{definition}
    Define the $k$\textsuperscript{th} \new{integral hook sum} $s_k$ (\Cref{intpcthooksums}) of a matrix $A = \left(a_{ij}\right) \in U_n$ as
    $$
        s_k=\sum_{i\leq k\leq j}{a_{ij}},\,1\leq k\leq n,
    $$ 
    and its \new{integral hook sum vector} as $\mathbf{ s}=\left(s_1,\,s_2,\,\ldots,\,s_n\right)$.

\begin{figure}[ht]
    \centering
    \small
\begin{equation*}
\begin{bmatrix}
\color{red} 0 & \color{red} 1 & \color{red} 0 & \color{red} 0& \color{red} 0 \\
  & 0 & 0 & 0& 1\\
 &   & 1 & 1& 0\\
&  &   & 2& 0\\
&  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
 0 & \color{red} 1 & \color{red} 0 & \color{red} 0& \color{red} 0 \\
  & \color{red}0 & \color{red}0 & \color{red}0& \color{red}1\\
 &   & 1 & 1& 0\\
&  &   & 2& 0\\
&  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 &  1 & \color{red} 0 & \color{red} 0& \color{red} 0 \\
  & 0 & \color{red}0 & \color{red}0& \color{red}1\\
 &   & \color{red}1 & \color{red}1& \color{red}0\\
&  &   & 2& 0\\
&  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 &  1 &  0 & \color{red} 0& \color{red} 0 \\
  & 0 & 0 & \color{red}0& \color{red}1\\
 &   & 1 & \color{red}1& \color{red}0\\
&  &   & \color{red}2&\color{red} 0\\
&  &  &  & 4
\end{bmatrix}
\begin{bmatrix}
0 &  1 &  0 &  0& \color{red} 0 \\
  & 0 & 0 & 0& \color{red}1\\
 &   & 1 & 1& \color{red}0\\
&  &   & 2&\color{red} 0\\
&  &  &  & \color{red}4
\end{bmatrix}
\end{equation*}
\caption{We add the \textcolor{red}{red} entries to get $\mathbf{s}=\left(1,\,2,\,3,\,4,\,5\right)$}
    \label{intpcthooksums}
\end{figure}
\end{definition}

\begin{definition}
    The set of all $U_n$ matrices with a given integral hook sum vector $\mathbf{s}$ is called the set of \new{(generalized) integral Tesler matrices} and denoted as $\T\left[\mathbf{s}\right]$ with $T\left[\mathbf{s}\right]=\left|\T\left[\mathbf{s}\right]\right|$.
\end{definition}

Now, notice that the matrix on \Cref{intpcthooksums} is precisely the one from \Cref{pcthooksums} after applying $I$. The map sent a matrix from $\T\left(1,\,1,\,1,\,1,\,1\right)$ to one from $\T\left[1,\,2,\,3,\,4,\,5\right]$. A general statement can be made.

\begin{proposition}
$I\left(\T\left(h_1,\,h_2,\,\dots,\,h_n\right)\right)=\T\left[h_1,\,h_1+h_2,\,\dots,\,\sum_{k=1}^n h_k\right].$
\end{proposition}
\begin{proof}
    Let $A\in \T\left[s_1,\,s_2,\,\dots,\,s_n\right]$, then observe $s_k-s_{k-1}= \sum_{i\geq k} {A_{ki}}-\sum_{i< k} A_{i(k-1)}$. This, in turn, is precisely $\sum_{i\geq k} {I^{-1}\left(A\right)_{ki}}-\sum_{i< k} I^{-1}\left(A\right)_{ik}$, which is $h_k$ for $I^{-1}\left(A\right)$.
\end{proof}

\begin{example}
    The integral version of $\mathbf{h}=\mathbf{1}^n$ is $\mathbf{s}=\left(1,\,2,\,\ldots,\,n\right)$ and so the regular Tesler matrices from \Cref{Tex} correspond to the integral Tesler matrices $\T\left[1,\,2\right]$:
\begin{align*}
\begin{bmatrix}
0 & 1 \\
& 1 
\end{bmatrix}\;\;\textsc{and}\;\;
\begin{bmatrix}
1 & 0 \\
& 2 
\end{bmatrix}.
\end{align*}
\end{example}

\subsection{Asymptotics analysis via integral Tesler matrices and partitions}
\label{ss:Asymptotics}
We will analyze the asymptotic behavior of the functions defined as $T\left[\mathbf{s}\left(n\right)\right]: \Z^+\to\Z^+$, where $\mathbf{s}\left(n\right)$ is a vector-valued function. Due to exponential growth, it is only practical to study $\ln T\left[\mathbf{s}\left(n\right)\right]$, and so from now on we shall use the notation $L\left[\mathbf{s}\right]=\ln T\left[\mathbf{s}\right]$.

It is natural that the asymptotics differ considerably as one changes $\mathbf{s}\left(n\right)$. For example, it is easy to verify that $L\left[\mathbf{1}^n\right]=\left(n-1\right)\ln2$. 
More importantly for us, it was shown (by a clever appeal to the Morris identity) in \cite{zeilberger1998proof} that $L\left(1,2,\ldots,n\right)=\sum_{k=1}^n{\ln C_k}\sim n^2\ln2,$ where $C_k$ denotes $k$\textsuperscript{th} Catalan number $\frac{1}{k+1}\binom{2k}{k}$.
Based on this, Igor Pak raised the question whether $L\left(\mathbf{1}^n\right)\asymp n^2$ as well (for asymptotics symbols $\prec,\preccurlyeq,\precsim,\asymp$ see \hyperref[app]{Appendix}).

In this paper we present the answer to the question above and also the general result for a wide range of integral hook sum sequences. That is achieved via breaking complicated $\mathbf{s}\left(n\right)$ into integral hook sums of much simpler structure. Those are the basic building blocks (atoms) of our analysis, and therefore, sufficient time should be devoted to studying them. To denote those, we shall introduce a binary operation $\times$ that maps integer pairs from $\Z^+\times\Z^+$ into vectors 
$\underbrace{\left(a,\,a,\,\ldots,\,a\right)}_{b\text{ times}} = a\times b \;\in\;\left(\Z^+\right)^b.$

The fundamental integral hook sums are then $\mathbf{s}\left(n\right) = H\left(n\right) \times W\left(n\right)$, where we have functions $H,W: \Z^+ \to \Z^+$. Now the aim is to build a small theory behind the asymptotics of
$L\left[H\left(n\right) \times W\left(n\right)\right]$, or, as it is convenient to omit indices (the implicit index $n$), $L\left[H\times W\right]$.
For the best previous analysis, see \cite{morales}.

% \subsection{Asymptotics analysis via integral Tesler matrices and partitions}

Note that having a fixed integral hook sum is too strict a constraint to do any counting. Then comes the idea of \new{representation}. We construct a superset of $\T\left[H\times W\right]$ that is easier to work with, but which, under logarithm, has the same asymptotics. In this case, we declare that $\T\left[H\times W\right]$ represents this superset. 

\begin{definition}\label{universe} The disjoint union of all $\T\left[s_1,\,s_2,\,\dots,\,s_n\right]$ such that $\sum s_k = m$ is called \new{integral Tesler matrices universe}, and denoted as $\T\left\{m,\,n\right\}$. Also, $L\left\{m,\, n\right\}=\ln T\left\{m,\, n\right\}.$
\end{definition}

We will prove later that $L\left[H\times W\right]\sim L\left\{HW,\,W\right\}$. For now, we shall think of ways to enumerate $\T\left\{m,\,n\right\}$. Here, we face the opposite problem to the one before: the defining restriction of the new set is too loose. The immediate idea is to choose a good representative of the universe, which is, in fact, pivotal. This motivates the next definition.

\begin{definition}
    Define the $k$\textsuperscript{th} \new{(weighted) row sum} $r_k$ of a matrix $A = \left(a_{ij}\right) \in U_n$ as
    $$
        r_k= a_{kk} +2a_{k\left(k+1\right)}+\dots+(n-k+1)a_{kn} = \sum_{j=k}^n{\left(j-k+1\right)a_{kj}}
    $$ 
    and its \new{(weighted) row sum vector} as $\mathbf{ r}=\left(r_1,\,r_2,\,\ldots,\,r_n\right)$.
\end{definition}
\begin{definition}
    The set of all $U_n$ matrices with fixed row sum vector $\mathbf{r}$ shall be called the set of \new{row Tesler matrices} and denoted as $\T_r\left(\mathbf{r}\right)$. As before, $T_r\left(\mathbf{r}\right)=\left|\T_r\left(\mathbf{r}\right)\right|$, and $  L_r\left(\mathbf{r}\right) = \ln T_r\left(\mathbf{r}\right).$
\end{definition}
\begin{example}
    Here are the only two matrices in $\T_r\left(2,\,1\right)$:
\begin{align*}
\begin{bmatrix}
0 & 1 \\
& 1 
\end{bmatrix}\;\;\textsc{and}\;\;
\begin{bmatrix}
2 & 0 \\
& 1 
\end{bmatrix}.
\end{align*}
\end{example}


We shall elaborate on the choice of definition. Consider the sum of all integral hook sums $\sum s_k$ of some $A = (a_{ij})\in U_n$ and count the coefficients before each $a_{ij}$ in it (see \Cref{FigRowSums}). As a result, those coefficients in each row perfectly match the ones from the definition of $r_i$. Therefore, $\sum r_k =\sum s_k$, and so the following proposition holds.
\begin{figure}[ht]
    \centering
    \small
\begin{equation*}
\begin{bmatrix}
\color{red} 1 & \color{red} 1 & \color{red} 1 & \color{red} 1& \color{red} 1 \\
  & 0 & 0 & 0& 0\\
 &   & 0 & 0& 0\\
&  &   & 0& 0\\
&  &  &  & 0
\end{bmatrix}
\begin{bmatrix}
 1 & \color{red} 2 & \color{red} 2 & \color{red} 2& \color{red} 2 \\
  & \color{red}1 & \color{red}1 & \color{red}1& \color{red}1\\
 &   & 0 & 0& 0\\
&  &   & 0& 0\\
&  &  &  & 0
\end{bmatrix}
\begin{bmatrix}
1 &  2 & \color{red} 3 & \color{red} 3& \color{red} 3 \\
  & 1 & \color{red}2 & \color{red}2& \color{red}2\\
 &   & \color{red}1 & \color{red}1& \color{red}1\\
&  &   & 0& 0\\
&  &  &  & 0
\end{bmatrix}
\begin{bmatrix}
1 &  2 &  3 & \color{red} 4& \color{red} 4 \\
  & 1 & 2 & \color{red}3& \color{red}3\\
 &   & 1 & \color{red}2& \color{red}2\\
&  &   & \color{red}1&\color{red} 1\\
&  &  &  & 0
\end{bmatrix}
\begin{bmatrix}
1 &  2 &  3 &  4& \color{red} 5 \\
  & 1 & 2 & 3& \color{red}4\\
 &   & 1 & 2& \color{red}3\\
&  &   & 1&\color{red} 2\\
&  &  &  & \color{red}1
\end{bmatrix}
\end{equation*}
\caption{Counting \textcolor{red}{coefficients} before every $a_{ij}$ in partial sums $s_1+s_2+\dots+s_k$}
\label{FigRowSums}
\end{figure}
\begin{proposition}\label{prop_r-s} Integral Tesler matrices universe is a universal set for $\T_r$:
    $$\T\left\{m,\,n\right\}=\bigcup_{r_1+r_2+\dots+r_n=m} \T_r\left(r_1,\,r_2,\,\dots,\,r_n\right).$$
\end{proposition}

The proposition above importantly enables the use of the following trick: if we have a decomposition of a set $A=\cup_{k=1}^{m} A_k$, then provided $\ln m\prec \ln\left|A\right|$, we have $\ln\max\left|A_k\right|\leq\ln\left|A\right|\sim \ln\max\left|A_k\right|+\ln m$, and so $\ln\left|A\right|\sim \ln\max\left|A_k\right|$. Due to extensive use, we shortcut by saying that the decomposition \new{induces the representative}
$\ln\max\left|A_k\right|$.

So far, we have united all the integral Tesler matrices into a superset, and then split it into pieces of the same number as before unification. However, amazingly, those new pieces are extremely easy to count.
\begin{proposition}\label{prop_r-par} Let $p\left(n;\;l,\,r\right)$ denote the number of unordered partitions of $n$ in which the size of each part $s$ satisfies $l\leq s\leq r$, then $$T_r\left(r_1,\,r_2,\,\dots,\,r_n\right)=\prod_{k=1}^n p\left(r_k;\;1;\;n-k+1\right).$$
\end{proposition}

Here we first find an unforeseen connection between Tesler matrices and integer partitions. The following fact will prove crucial.

\begin{lemma}\label{partition_lemma}
    Let $l\left(n\right)\prec \sqrt n\prec r\left(n\right)$, then
    $$\ln p\left(n;\;l,\,r\right)\sim \ln p\left(n;\;1,\,n\right)\sim\sqrt{\frac{2}{3}}\pi\sqrt{n}.$$
\end{lemma}
\begin{proof}
    The second equivalence is the famous Hardy-Ramanujan formula, so we derive the first one. Any partition is first partitioning $n$ into three numbers $n_1,\,n_2,\,n_3$ and then partitioning them into parts of size from $1$ to $l-1$, $l$ to $r$, and $r+1$ to $n$ respectively. 
    This leads to the following bounds: $$\ln p\left(n;l,r\right) \leq \ln p\left(n;1,n\right)\leq  \ln\binom{n+2}{2}+\ln p\left(n;1,l-1\right) +\ln p\left(n;l,r\right) +\ln p(n;r+1,n).$$ 
    
    It follows from the asymptotic formula of \cite{partition} that all the summands to the right are asymptotically negligible compared to $\sqrt{\frac{2}{3}}\pi\sqrt{n}$, except for $\ln p\left(n;\;l,\,r\right)$.
\end{proof}

It is now high time to move back to $L\left[H\left(n\right) \times W\left(n\right)\right]$. We shall first do so for $W^2\succ H$.
\begin{theorem}\label{Lemma_firstset}
    Let $H,\,W:\mathbb{Z}^+\to \mathbb{Z}^+$ such that $W^2\succ H\succ 1$. Then
    $$L\left\{HW,\,W\right\}\sim L_r\left(H\times W\right)=L_r\underbrace{\left(H(n),\,H(n),\,\dots,\,H(n)\right)}_{W(n)\text{ times}}\sim \sqrt{\frac{2}{3}}\pi W\sqrt{H}.$$
\end{theorem}
\begin{proof} We combine \Cref{prop_r-par} with \Cref{partition_lemma} (we can use it as $1\prec \sqrt{H}\prec W$) to obtain $L_r\left(H\times W\right)=\sum_{k=1}^{W}\ln p\left(H;\;1,\,W-k+1\right)\sim W\cdot \sqrt{\frac{2}{3}}\pi\sqrt{H}.$

What is left is to obtain a strict upper-bound for $L\left\{HW,\,W\right\}$. \Cref{prop_r-s} induces the representative 
 $L\left\{HW,\,W\right\} \sim\max L_r\left(\mathbf{r}\right)$, which we can compute as
$$\max \sum_{k=1}^W \ln p\left(r_k;\,1,\, W - k + 1\right)  \precsim \sqrt{\frac{2}{3}}\pi \cdot\max \sum_{k=1}^W \sqrt{r_k}= \sqrt{\frac{2}{3}}\pi W \sqrt{H},$$ since for fixed $\sum_{k=1}^W r_k$, the sum $\sum_{k=1}^W \sqrt r_k$ is maximized under $r_1=r_2=\dots=r_W$.
\end{proof}

We shall now show that $\T\left[H\times W\right]$ is indeed a representative of $\T\left\{HW\times W\right\}.$

\begin{theorem}\label{Theo}
    Let $H,\,W:\mathbb{Z}^+\to \mathbb{Z}^+$ such that $W^2\succ H\succ 1$. Then
    $$L\left[H\times W\right]\sim \sqrt{\frac{2}{3}}\pi W\sqrt{H}.$$
\end{theorem}
\begin{proof}
    We shall ``carve'' (almost) a subset of $\T\left[H\times W\right]$ from $\T\left\{HW,\,W\right\}$ by imposing several restrictions loose enough though to preserve the asymptotics.

    We want all $s_k$ equal. However, the corresponding summation regions differ significantly, so we need to define a restriction first. Let $l\sim\alpha^{-1}\sqrt{H},\,r\sim\alpha\sqrt{H}$ be some functions, where $1\prec \alpha\prec \min\left(W/\sqrt{H},\,\ln H\right)$. Then let $\mathcal{A}\subseteq\T\left\{HW,\,W\right\}$ be the subset of matrices with non-zero entries $a_{ij}$ having either $j> W-r+1$ or $l\leq j-i+1\leq r$. Now we have non-zero elements only on portions of consecutive entries in each row (except for the last $r-1$ ones), which we shall call \new{segments} (see \Cref{fig_entries}).  

    \begin{figure}[ht]
        \centering
        \small
    \begin{tikzpicture}[scale=0.27]
    \draw[thick] (1, 10)--(1, 10.5) node[above] {$l$};
    \draw[thick] (2, 10)--(2, 10.5) node[above] {$r$};
    \draw[<->] (10.7,0) -- (10.7,10);
    \node at (11.7,5) {$W$};
    \draw [decorate,decoration={brace,amplitude=2}]
  (7.6,-.3) -- (7.6,1.9) node[midway,xshift=-17]{$r-1$};
  \draw[dotted] (7.6,-.3)--(10,-.3);
    \draw[dotted] (7.6,1.9)--(8,1.9);
  % вершины большого треугольника
  \coordinate (A) at (0,10);
  \coordinate (B) at (10,10);
  \coordinate (C) at (10,0);
  \coordinate (C1) at (9.95,.15);
  \coordinate (G) at (8.1,2);

  % точки пересечения линий ℓ и r с AB и BC
  \coordinate (Ltop) at (1,9.95);
  \coordinate (Lbot) at (9.95,1);
  \coordinate (Rtop) at (2,9.95);
  \coordinate (Rbot) at (9.95,2);

  % точка D для красного треугольника
  \coordinate (D) at ($(A)!{(1-0.2)/1}!(C)$);

  % сам треугольник
  \draw[very thick] (A) -- (B) -- (C) -- cycle;

  % заштрихованная голубая полоса
  \fill[blue!30] (Ltop) -- (Rtop) -- (Rbot) -- (Lbot) -- cycle;
  % синие границы — просто линии без стрелок и меток
  % \draw[blue,thick] (Ltop) -- (Lbot);
  % \draw[blue,thick] (Rtop) -- (Rbot);

  % маленький красный треугольник
  \fill[red!50] (C1) -- (Rbot) -- (G) -- cycle;
  % \draw[red,thick] (D) -- (Lbot) -- (C) -- cycle;
\end{tikzpicture}
        \caption{\centering Consecutive non-zero entries in each row (segments) are colored \textcolor{blue}{blue}, the last rows without segments are colored \textcolor{red}{red}, zero entries are colored white.}
        \label{fig_entries}
    \end{figure}

    We would like all segments to have equal entries, equalizing all $s_k$. However, this drastically reduces cardinality, so we need to adjust the restrictions. We divide every segment into consecutive \new{blocks} of equal length $\gamma\sim\alpha^2\ln H$ so that there are $\beta=\left(r-l+1\right)/\gamma\sim \alpha^{-1}\sqrt{H}/\ln H$ blocks in any segment. Let $m_{ik}$ denote the weighted sum $\sum{\left(j-i+1\right)a_{ij}}$ over the $k$\textsuperscript{th} block of the $i$\textsuperscript{th} segment. Those can be put in block sum matrix $M=\left[m_{ik}\right]$. Our idea now is to equate sequences $\left(m_{ik}\right)_{1\leq k\leq \beta}$ over all $1\leq i\leq W-r+1$.

    This would equate all the row sums, so we shall require $\mathcal{A}\subseteq\T_r\left(H\times W\right)$. From \Cref{prop_r-par} and \Cref{partition_lemma} we have $\ln\left|\mathcal A\right|\sim\sqrt{2/3}\pi W\sqrt{H}$.
    Now, denote by  $\mathcal{A}_M\subseteq\mathcal{A}$ the set of matrices from $\mathcal{A}$ with block sum matrix $M=\left[m_{ik}\right]$. Then, since each partition in $p\left(H;\; l,\, r\right)$ corresponds to a sequence of weighted block sums, we choose the sequence $m'_1,\,m'_2,\,\dots,\,m'_\beta$ associated with the largest number of partitions. From it we construct a matrix $M'=\left[m_{ik}\right],\,m_{ik}=m'_k$. Clearly, $\left|\mathcal{A}_{M'}\right|=\max\left|\mathcal{A}_M\right|$, and so it is the induced representative of $\mathcal{A}.$ 
    
    Some basic computations show $s_k\sim H,\,r\leq k\leq W-r+1$ for the matrices of $\mathcal{A}_{M'}$. Now, as we want to connect this to $\T\left[H\times W\right]$, we shall restrict $\mathcal{A}_{M'}$ to just one integral hook sum vector (as for now it contains matrices with different vectors). This is easily done, as we can decompose $\mathcal{A}_{M'}$ into subsets with fixed $\mathbf{s}$, and induce some representative $\mathcal{B}\subseteq \mathcal{A}_{M'}$.

    Finally, our argument runs as follows:
    the above holds for every $H$ as long as $W^2\succ H\succ 1$, so we can replace every $H$ by some $qH,\, q<1$ to get $\mathcal B$ with $s_k\sim qH,\,r\leq k\leq W-r+1$. For sufficiently large $n$, we, in fact, have $s_k\left(n\right)\leq H\left(n\right)$, and so it is easily obtained $L\left[H\times W\right]\succsim \ln\left|\mathcal B\right|$, culminating in $L\left[H\times W\right]\succsim \sqrt{\frac{2}{3}}\pi W\sqrt{qH}\xrightarrow{q \to 1} \sqrt{\frac{2}{3}}\pi W\sqrt{H}. 
    $\qedhere
\end{proof}

As it turns out, all of this can be much easier done for $H\succ W^2.$

\begin{theorem}\label{theo2}
    Let $H,\,W:\mathbb{Z}^+\to \mathbb{Z}^+$ such that $H\succ W^2\succ 1$. Then
    $$L\left[H\times W\right]\sim L\left\{HW,\,W\right\}\sim L_r\left(H\times W\right)\sim W^2 \ln\left(\frac{\sqrt{H}}{W}\right).$$
\end{theorem}
\begin{proof} The proof is analogous to \Cref{Lemma_firstset}. Thus we proceed with
    $$L_r\left[H\times W\right]=  \sum_{k=1}^W{\ln p\left(H;\;1,\,k\right)}\sim \sum_{k=1}^W{k\ln\left(\frac{H}{k^2}\right)}\sim \int_{1}^W{k\ln\left(\frac{H}{k^2}\right)dk}\sim W^2 \ln\left(\frac{\sqrt{H}}{W}\right),$$     where we use the $p\left(H;\;1,\,k\right)$ asymptotics from \cite{partition}. Now, we induce the representative $L\left\{HW,\,W\right\} \sim  \max L_r\left(\mathbf{r}\right)$, which we shall expand utilizing the same partition asymptotic formula as
    $$\max L_r\left(\mathbf{r}\right)=\max\sum_{k=1}^W{\ln p\left(r_k;\;1,\,W-k+1\right)}\sim \max\sum_{k=1}^W{k\ln\left(\frac{r_k}{k^2}\right)}.$$
    As this is maximized under $r_k=2kH/W$, we can write  
    $$L\left\{HW,\,W\right\}\precsim \sum_{k=1}^W{k\ln\left(\frac{2H}{kW}\right)}\sim\int_1^W{k\ln\left(\frac{2H}{kW}\right)\,dk}\sim W^2\ln\left(\frac{\sqrt{H}}{W}\right).$$
    Finally, we shall provide a sharp lower-bound for $L\left[H\times W\right]$. To do this, consider all the integral Tesler matrices with sum over all entries being $H$ to derive \[L\left[H\times W\right]\geq \ln \binom{H+W\left(W-1\right)/2-1}{H}\sim W^2\ln\left(\frac{\sqrt{H}}{W}\right).\qedhere
    \]
\end{proof}
While the asymptotics of $T\left[H\times W\right]$ has been investigated, with an improved lower bound presented in \cite[Theorem 1.1]{morales}, it was done so for a constant $H\left(n\right)=H_0$ and $W\left(n\right)\to\infty$. Here, in contrast, both $H,\,W\to\infty$, allowing us to use asymptotics for the number of partitions to produce such sharp estimates.
\begin{proposition}\label{theo3}
    Let $H,\,W:\mathbb{Z}^+\to \mathbb{Z}^+$ such that $H\asymp W^2\succ 1$. Then
    $$L\left[H\times W\right]\asymp L\left\{HW,\,W\right\}\asymp L_r\left(H\times W\right)\asymp H.$$
\end{proposition}
\begin{proof}
    Let $W_1\prec \sqrt H\prec W_2$, then we can bound all these functions by the previous theorems as
    $ W_1^2 \ln\left(\frac{\sqrt{H}}{W_1}\right)\precsim L\left[H\times W\right],\, L\left\{HW,\,W\right\},\, L_r\left(H\times W\right)\precsim \sqrt{\frac{2}{3}}\pi W_2\sqrt{H}.$
    If we now let $W_1,\,W_2\to \sqrt{H}$, we would get the desired result.
\end{proof}
\section{Kostant pictures and their asymptotics}\label{ovs_pid}
\subsection{Kostant pictures and height diagrams}

As claimed in the introduction to the question of asymptotics analysis, we shall connect a large class of $\mathbf{s}\left(n\right)$, including $\mathbf{s}\left(n\right)=\left(1,\,2,\,\dots,\,n\right)$, to the atomic $\mathbf{s}\left(n\right)=H\left(n\right)\times W\left(n\right)$ in this section. Our chief tool in doing so will be \new{Kostant pictures}.

It has been shown \cite{M_sz_ros_2016} that the number of generalized Tesler matrices is connected to the type $A_n$ \new{Kostant partition function} which arises in representation theory. The combinatorial model below is due to \cite{AnKo}.

\begin{definition}  
    A \new{Kostant picture} is a diagram consisting of $n\in\Z^+$ ordered vertices and loops around consecutive vertices. Denote the number of loops enclosing vertices $i,\,i+1,\,\dots j+1$ as $a_{ij}$. Additionally, let $\left(e_1,\,e_2,\,\dots,\, e_{n+1}\right)$ be the standard orthonormal vector basis of $\R^{n+1}$, and define $\alpha_{ij}=e_i-e_{j+1}$. Then, the vector $\varnu = \sum a_{ij}\alpha_{ij}$ is called the \new{weight} of this Kostant picture. The set of all possible Kostant pictures for a given $\varnu$ is denoted by $\K\left(\varnu\right)$, while their number $\KPF\left(\varnu\right)$ defines the \new{Kostant partition function}.
\end{definition}

\begin{lemma*}[M\'esz\'aros, Morales, Rhoades \cite{M_sz_ros_2016}]
    $T\left(\varnu\right) = \KPF\left(\varnu\right).$
\end{lemma*}

As we have already seen, it is beneficial to ``integrate'' the hook sum vector, and so we shall do the same with weight. To each weight $\varnu=\left(\varnu_1,\,\varnu_2,\,\dots,\,\varnu_{n+1}\right)$ we associate the height $\psi=\left(\varnu_1,\,\varnu_1+\varnu_2,\,\ldots,\,\sum_{k=1}^{n}{\varnu_k}\right)$. This way, $\psi_k$ denotes the number of loops containing vertex $k$. We denote the set of all Kostant pictures with height $\psi$ by $\K\left[\psi\right]$ (analogously to how we defined $\T\left[\mathbf{s}\right]$), and we already know that $\left|\K\left[\psi\right]\right|=T\left[\psi\right]$. We shall stick to this $T$ notation as we have already used it extensively.

Height $\psi$ can be visually represented by its \new{height diagram}, which is simply the graph of $y=\psi\left(x\right)$ in the $xy$-plane. It is a discrete function that is zero on its entire domain except for a finite number of arguments (a subset of $\left\{1,\,2,\,\dots,\,n\right\}$). That being said, we will draw it continuously in the next sections for the sake of clarity. 
The utility of height diagrams lies in the following easy observation. 

\begin{proposition}\label{prop:nugekpfge}
    Let $\psi_1,\,\psi_2$ be two heights such that $\psi_1\left(x\right) \geq \psi_2\left(x+a\right),\,x\in\mathbb Z$ holds for some integer $a$. 
    Then $\KPF\left[\psi_1\right] \geq \KPF\left[\psi_2\right]$.
\end{proposition}
\begin{proof}
    Consider the subset of $\K\left[\psi_1\right]$, which contains all the Kostant pictures with the number of loops $\alpha_{ii}$ at least $\psi_1\left(i\right) - \psi_2\left(i+a\right)$ for all $i$. Then there is an obvious bijection between this set and $\K\left[\psi_2\right]$: given a picture, for all $i$ delete $\psi_1\left(i\right) - \psi_2\left(i+a\right)$ loops of the type $\alpha_{ii}$ and    change each $\alpha _{ij}$ by $\alpha_{i-a,\,j-a}$ . 
\end{proof}

With no work at all we can already use height diagrams to connect a known and an unknown asymptotics to the atomic ones.

\begin{proposition}\label{BasicHeight}
    $L\left(\mathbf{1}^n\right)\asymp L\left[n\times n\right]\;\;\textsc{and}\;\;L\left(1,\,2,\,\ldots,\,n\right)\asymp L\left[n^2\times n\right].$
\end{proposition}
% 
\begin{proof}
    Note that weight $\mathbf 1^n$ corresponds to the height $\psi\left(x\right)=x$, and so by looking at \Cref{examplehight1} we can deduce (recall \Cref{prop:nugekpfge})
   $L\left[\frac{n}{2}\times \frac{n}{2}\right]\leq L\left(\mathbf 1 ^n\right)\leq L\left[n\times n\right].$
   As $L\left[\frac{n}{2}\times \frac{n}{2}\right]\asymp L\left[n\times n\right]$, we get the desired result. The same can be stated for the second proposition (see \Cref{examplehight2}).
\begin{figure}[ht]
\centering
\small
\begin{minipage}[b]{0.4\linewidth}
\centering
\begin{tikzpicture}
\draw[->] (0, 0) -- (4.3, 0);
\draw[->] (0, 0) -- (0,4.3);
\draw[-] (4,4)--(0, 0) node[left] {$0$};
\draw[dotted]  (2, 2)--(0,2) node[left] {$\frac{n}{2}$};
\draw[-]  (4, 2)--(2,2);
\draw[-] (2, 2)--(2, 0) node[below] {$\frac{n}{2}$};
\draw[-] (4, 4)--(4, 0) node[below] {$n$};
\draw[-] (4, 4)--(0, 4) node[left] {$n$};
\end{tikzpicture}
\caption{$L\left(\mathbf{1}^n\right)\asymp L\left[n\times n\right]$}
\label{examplehight1}
\end{minipage}\hfill
\hfill
\begin{minipage}[b]{0.55\linewidth}
\centering
\begin{tikzpicture}
\draw[->] (0, 0) -- (3.13, 0);
\draw[->] (0, 0) -- (0,4.3);
\node[left] at (0, 0)  {$0$};
\draw[domain=0:2.83, smooth, variable=\x] plot ({\x}, {\x*\x/2});
\draw[dotted]  (1.4, 1)--(0, 1) node[left] {$\frac{1}{2}\left(\frac{n^2}{4}+\frac{n}{2}\right)$};
\draw[-]  (2.83, 1)--(1.4, 1);
\draw[-] (1.4, 1)--(1.4, 0) node[below] {$\frac{n}{2}$};
\draw[-] (2.83, 1)--(2.83, 0) node[below] {$n$};
\draw[-] (2.83, 4)--(2.83, 0);
\draw[-] (2.83, 4)--(0, 4) node[left] {$\frac{1}{2}\left(n^2+n\right)$};
\end{tikzpicture}
\caption{$L\left(1,\,2,\,\ldots,\, n\right)\asymp L\left[n^2\times n\right]$}
\label{examplehight2}
\end{minipage}
\caption{Basic height diagrams usage}
\end{figure}
\end{proof}
% 

\subsection{Asymptotics analysis via colorings and height decomposition}


In the previous section, asymptotics for the type $\psi=H\times W$ was derived. Now, we shall analyze the type $\psi\left(n\right)=\left(a_1,\,a_2,\,\dots,\,a_n\right)$ for some fixed integer sequence $\left(a_n\right)_{n\geq 1}$. In other words, one obtains $\psi\left(n+1\right)$ by appending $a_{n+1}$ to $\psi\left(n\right)$. This will give us the asymptotic formula for the number of regular Tesler matrices on substituting $a_n=n$ (answering questions in \cite{morales, ONeill}).

Now we want a more accurate version of \Cref{BasicHeight}. So we can do \new{height decomposition} into pieces that can be well approximated with $\psi = H\times W$. For this, we introduce the following concepts.
    $\K^k\left[\psi\right]$ denotes the set of Kostant pictures with height $\psi$, in which every loop is colored with one of $\left\lfloor k\right\rfloor$ distinct colors. Also, $T^k\left[\psi\right]=\left|\K^k\left[\psi\right]\right|$ and $L^k\left[\psi\right]=\ln T^k\left[\psi\right].$
    $C_k\left[\psi\right]$ denotes the maximal number of colorings a Kostant picture from $\K\left[\psi\right]$ can have into $\left\lfloor k\right\rfloor$ distinct colors.
\begin{proposition}\label{prop111}
    $\max\left(T\left[\psi\right],\, C_k\left[\psi\right]\right)\leq T^k\left[\psi\right]\leq T\left[\psi\right]\cdot C_k\left[\psi\right].$
\end{proposition}
\begin{theorem}[Decomposition Theorem]\label{DecProp}
    Let $\psi=\left(\psi_1,\,\psi_2,\,\dots,\,\psi_{kW}\right)$ be some height. \\ Define sequences $\phi_t=\left(\psi_{\left(t-1\right)W+i}\right)_{1\leq i\leq W},\,1\leq t\leq k$ and 
    $$S=\prod_{i=1}^k T\left[\psi_{\left(i-1\right)W+1},\,\psi_{\left(i-1\right)W+2},\,\dots,\,\psi_{iW}\right].$$ 
    Then we can lower-bound and upper-bound $T\left[\psi\right]$ by     $$T\left[\psi\right]\geq \max\left(S,\,T^{\frac{W^2}{4}}\left[\min\left(\min \phi_1,\,\min \phi_2\right),\,\dots,\,\min\left(\min \phi_{k-1},\,\min \phi_k\right)\right]\right),$$ 
    $$T\left[\psi\right]\leq S\cdot T^{W^2}\left[\min\left(\max \phi_1,\,\max \phi_2\right),\,\dots,\,\min\left(\max \phi_{k-1},\,\max \phi_k\right)\right].$$
\end{theorem}
% 
\begin{proof}
 % (\Cref{HDecompose})
   Decompose $\psi$ into $k$ blocks of length $W$ each. If we ignore all possible loops $\alpha_{ij}$ that are contained within several such divisions, we have $T\left[\psi\right]\geq S$.
   
   Now, consider all such missing $\alpha_{ij}$ as determined by two parameters: the set of divisions the loop covers and the relative positions of its ends in the first and the last divisions. Note that in order to get a lower-bound, this situation can be modeled by 
   $$\K^{\left(W/2\right)^2}\left[\min\left(\min \phi_1,\,\min \phi_2\right),\,\min\left(\min \phi_2,\,\min \phi_3\right),\,\dots,\,\min\left(\min \phi_{k-1},\,\min \phi_k\right)\right],$$
   as we have $k-1$ division lines to intersect, and at least $\frac{W}{2}$ possible choices per loop end so that the loops with disjoint division sets do not cross. To get an upper bound, $W$ possible choices per end should be allowed and all $\min$ replaced with $\max$.
% \begin{figure}[ht]
% \centering
% \small
% \begin{tikzpicture}
% \draw[->] (0, 0) -- (12.3, 0);
% \draw[->] (0, 0) -- (0,4.3);
% \node[left] (0, 0)  {$0$};
% \draw[-] (12,4)--(0, 4) node[left] {$\psi\left(x\right)$};
% \draw[-] (12,4)--(12, 0) node[below] {$kW$};
% \draw[-] (2,4)--(2, 0) node[below] {$W$};
% \draw[-] (4,4)--(4, 0) node[below] {$2W$};
% \draw[-] (6,4)--(6, 0) node[below] {$3W$};
% \node at (9, 2) {$\cdots$};
% \draw[thick,  double] (1.2,0.3)--(11.5, 0.3);
% \draw[thick, dashed] (3.4,.5)--(6.5, .5);
% \draw[thick, densely dotted] (0.3,.5)--(2.7, .5);
% \draw[ultra thin] (2.6,.7)--(6.2, .7);
% \end{tikzpicture}
% \caption{Decomposition of $T\left[\psi\right]$}
% \label{HDecompose}
% \end{figure}
\end{proof}

For this theorem to be utilized we shall first have some asymptotics of $T^k\left[\psi\right]$, or alternatively, $C_k\left[\psi\right]$.

\begin{lemma}
    Let $\psi,\,k:\Z^+\to\Z^+$ be two function such that $k\succ \psi^2\succ 1$, then 
    $$\ln C_k\left[\psi\left(1\right),\,\psi\left(2\right),\,\ldots,\,\psi\left(n\right)\right]\ \sim\sum_{i=1}^n \psi\left(i\right)\ln\left(\frac{k}{\psi\left(i\right)}\right).$$
\end{lemma}
\begin{proof}
    Let $A=\left[a_{ij}\right]\in\T\left[\psi\left(1\right),\,\psi\left(2\right),\,\ldots,\,\psi\left(n\right)\right]$ correspond to the Kostant picture that maximizes the number of colorings into $k$ distinct colors, in other words, $$C_k\left[\psi\left(1\right),\,\psi\left(2\right),\,\ldots,\,\psi\left(n\right)\right]=\prod_{v\in \left[a_{ij}\right]}{\binom{v+k-1}{k-1}}.$$
    
    Note that in every row of $A$ we have entry values in descending order since otherwise we could permute values to get the same value of the product above, while lowering some integral hook sums. Now, let $a=a_{ii},\,b=a_{i\left(i+1\right)},\,c=a_{\left(i+1\right)\left(i+1\right)}$. In the sum of integral hook sums $\sum s_i$ those entries are counted as $a+2b+c$ (see \Cref{FigRowSums}). So, if we denote $\text{sum}=a+2b+c$, then we shall consider maximization of $$\binom{a+k-1}{k-1}\cdot\binom{c+k-1}{k-1}\cdot\binom{\frac{\text{sum}-a-c}{2}+k-1}{k-1}.$$

    The expression above is maximized when $b\sim a^2/k\sim c^2/k\prec 1$, and so we can assume values only on the main diagonal, which leads to
\[\ln C_k\left[\psi\left(1\right),\,\psi\left(2\right),\,\ldots,\,\psi\left(n\right)\right]\sim \sum_{i=1}^n \ln \binom{\psi\left(i\right)+k-1}{k-1}\sim \sum_{i=1}^n \psi\left(i\right)\ln\left(\frac{k}{\psi\left(i\right)}\right).\qedhere
    \]
\end{proof}
% \newpage
\begin{theorem}
\label{CorSum}
    Let $\psi:\Z^+\to\Z^+$ be a function such that $n^2\succ\psi\left(n\right)\succ 1$ and $\sum_{i=1}^n{\sqrt{\psi\left(i\right)}}\succ \psi\left(n\right)$. Then $$L\left[\psi\left(1\right),\,\psi\left(2\right),\,\ldots,\,\psi\left(n\right)\right]\sim \sqrt\frac{2}{3}\pi \sum_{i=1}^n{\sqrt{\psi\left(i\right)}}.$$
\end{theorem}
\begin{proof}
    Let $W$ be a function such that $n\succ W\left(n\right)\succ \sqrt{\psi\left(n\right)}$. We shall now use \Cref{DecProp} with $n=kW$. Consider $S$ first. We can easily bound it with \Cref{Theo} to derive
    $$ 
        \sqrt\frac{2}{3}\pi W\sum_{i=1}^{k-1}{\sqrt{\psi\left(iW\right)}}\precsim S\precsim  \sqrt\frac{2}{3}\pi W\sum_{i=1}^{k}{\sqrt{\psi\left(iW\right)}}.
    $$
   Since $\sum_{i=1}^n{\sqrt{\psi\left(i\right)}}\succ \psi\left(n\right)$, we can choose $W$ to satisfy $\sum_{i=1}^n{\sqrt{\psi\left(i\right)}}\succ  W\sqrt{\psi\left(n\right)}$, and so asymptotically equate the bounds above to get $S\sim \text{RHS}$ of the Theorem statement.
    
    It is only left to show that the bigger coloring term from \Cref{DecProp} is asymptotically insignificant. Use \Cref{prop111} and the previous Lemma to get the upper-bound for it:
    $L\left[\psi\left(2W\right),\,\psi\left(3W\right),\,\dots,\,\psi\left(n\right)\right]+\sum_{i=1}^{k-1} \psi\left(iW\right)\ln\left(\frac{W^2}{\psi\left(iW\right)}\right).$
    For the second summand, it is straightforward:
$\sum_{i=1}^n\sqrt{\psi\left(i\right)}\succ \psi\left(n\right)= k^2\cdot \frac{\psi\left(n\right)}{k^2}\geq k^2\ln\left(\frac{\psi\left(n\right)}{k^2}\right).$
    For the first one, first upper-bound it by $L\left[\psi\left(n\right)\times k\right]$, then 
    $\sum_{i=1}^n\sqrt{\psi\left(i\right)}\succ n\succ n\cdot\frac{\sqrt{\psi\left(n\right)}}{W}=k\sqrt{\psi\left(n\right)}.
    $ \qedhere
\end{proof}
% 
\begin{corollary}
\label{ATesler}
    $L\left(\mathbf{1}^n\right)\sim \sqrt\frac{8}{27}\pi n\sqrt{n}.$
\end{corollary}

For a proof use \Cref{CorSum} with $\sum_{i=1}^n\sqrt{i}\sim 2/3 n\sqrt{n}$. Note that the asymptotics presented in \Cref{ATesler} drastically improve the previous bound $L\left(\mathbf{1}^n\right) \geq \frac{n}{4} \ln^2n - \mathcal{O}(n\ln n)$ from \cite[Theorem 1.2]{morales}.

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way

\acknowledgements{We would like to thank: Anne Dranowski for mentoring us, her guidance, and support; Pavel Etingof, Slava Gerovitch, and Dmytro Matvieievskyi for the opportunity to conduct research; Nathan Williams, Alejandro Morales, and GaYee Park for their suggestions.}
\subsection*{Appendix: Asymptotics}
\label{app}
\begin{definition}
    As $x\to\infty$, we write $f\sim g$ when $f/g\to 1$; $f\prec g$ when $f/g\to0$;  $f\asymp g$ when $ag\leq f\leq bg$ for some $a,\, b\in\R^+$; $f\precsim g$ when $f\leq ag$ for every $a>1$ ($x\to\infty$ for every $a$ independently); and $f\preccurlyeq g$ when $f\leq ag$ for some $a\in\R^+$.
\end{definition}


\printbibliography
\end{document}