\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\underline{#1}}} \usepackage{enumerate,tikz,listings,array,longtable,url,hyperref} \usetikzlibrary{shapes,arrows,chains} \usepackage{pgf} \DeclareMathOperator{\OCH}{OCH} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{problem}[theorem]{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \begin{center} \vskip 1cm{\LARGE\bf Antichain Simplices} \vskip 1cm \large Benjamin Braun and Brian Davis \\ Department of Mathematics\\ University of Kentucky\\ Lexington, KY 40506-0027\\ USA \\ \href{mailto:benjamin.braun@uky.edu}{\tt benjamin.braun@uky.edu} \\ \href{mailto:brian.davis@uky.edu}{\tt brian.davis@uky.edu} \end{center} \vskip .2 in \begin{abstract} Associated with each lattice simplex $\Delta$ is a poset encoding the additive structure of lattice points in the fundamental parallelepiped for $\Delta$. When this poset is an antichain, we say $\Delta$ is antichain. For each partition $\lambda$ of $n$, we define a lattice simplex $\Delta_\lambda$ having one unimodular facet, and we investigate their associated posets. We give a number-theoretic characterization of the relations in these posets, as well as a simplified characterization in the case where each part of $\lambda$ is relatively prime to $n-1$. We use these characterizations to experimentally study $\Delta_\lambda$ for all partitions of $n$ with $n\leq 73$. Further, we experimentally study the prevalence of the antichain property among simplices with a restricted type of Hermite normal form, suggesting that the antichain property is common among simplices with this restriction. Finally, we explain how this work relates to Poincar\'e series for the semigroup algebra associated with $\Delta$, and we prove that this series is rational when $\Delta$ is antichain. \end{abstract} \newcommand{\C}{\mathbb{C}} \newcommand{\PP}{\mathrm{P}} \newcommand{\B}{\mathbb{B}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\fS}{\mathfrak{S}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\cone}{\mathrm{cone}} \newcommand{\conv}{\mathrm{conv}} \newcommand{\height}{\mathrm{height}} \newcommand{\im}{\mathrm{im}} \newcommand{\HH}{\mathcal{H}} \newcommand{\fpa}{\mathrm{FPA}} \newcommand{\pn}{\mathrm{part}} \newcommand{\RP}{\mathrm{RelPrime}} \newcommand{\rp}{\mathrm{relprime}} \newcommand{\rpac}{\mathrm{rpac}} \newcommand{\relprime}{\mathrm{relprime}} \newcommand{\ptn}{\mathrm{part}} \newcommand{\Rpac}{\mathrm{Rpac}} \newcommand{\Relprime}{\mathrm{Relprime}} \newcommand{\Ptn}{\mathrm{Part}} \def\v{{\boldsymbol v}} \def\t{{\boldsymbol t}} \newcommand{\A}{\mathcal{V}} \newcommand{\m}{\mathfrak{m}} \section{Introduction}\label{sec:intro} Given a lattice simplex $\Delta$, the structure of the lattice points in the fundamental parallelepiped of the cone over $\Delta$ reflects a wealth of arithmetic and combinatorial properties of $\Delta$. In this work, we study a partial order on these lattice points that encodes the additive relations among these points. Thus, our main results are primarily arithmetic in nature, and will hopefully be of interest to those working in areas where fundamental parallelepipeds play a role, e.g., Ehrhart theory, partition identities, coding theory, optimization, etc. Our motivation for this investigation comes from questions regarding rationality of Poincar\'e series for infinite graded resolutions of graded algebras, where it is of particular interest when the partial order associated with $\Delta$ has no relations. Thus, after developing our main results using number-theoretic techniques, we explain their algebraic implications. More precisely, in Section~\ref{sec:fpp} we define the fundamental parallelepiped poset $P(\Delta)$ associated with $\Delta$, where we say $\Delta$ is \emph{antichain} if $P(\Delta)\smallsetminus \{0\}$ has no relations. For each partition $\lambda$ of $n$, we define a lattice simplex $\Delta_\lambda$ having one unimodular facet, and we investigate the posets for these simplices in depth. In Theorem~\ref{thm:order} we give a number-theoretic characterization of the relations in $P(\Delta_\lambda)$, and in Corollary~\ref{cor:order} we give a simplified characterization in the case where each part of $\lambda$ is relatively prime to $n-1$. In Section~\ref{sec:experiments} we use these characterizations to generate empirical data, experimentally studying those $\Delta_\lambda$ for all partitions of $n$ with $n\leq 73$. These experiments reveal that a substantial fraction of those $\lambda$ satisfying the relatively prime condition appear to have $\Delta_\lambda$ that are antichain. Further, we experimentally study the prevalence of the antichain property among simplices with a restricted type of Hermite normal form, suggesting that the antichain property is common among simplices with this restriction. Finally, in Section~\ref{sec:algebraic} we explain the algebraic implications of our work to Poincar\'e series of semigroup algebras associated with $\Delta$. Specifically, we prove that if $\Delta$ is antichain, then the associated Poincar\'e series is rational. \section{Fundamental parallelepiped posets and their structure}\label{sec:fpp} \subsection{Lattice simplices and associated posets} Details regarding polytopes, cones, Hilbert bases, etc. as discussed in the following can be found in standard references~\cite{CCD,MillerSturmfels}. For a collection $\A=\{v_0,\dots,v_m\}$ of points in $\R^d$, we let $\conv(\A)$ denote the convex hull of $\A$. In the case that $m=d$ and the set $\A^\circ\,:=\,\{(v_1-v_0),\dots,(v_d-v_0)\}$ is a vector space basis of $\R^d$, then we call $\Delta := \conv(\A)$ a $d$-simplex. We call the $v_i$'s the {\emph{vertices}} of $\Delta$, and if each $v_i$ is an integer point, i.e., lies in $\Z^d$, we call $\Delta$ a {\emph{lattice simplex}}. We define the {\emph{conical hull}} of $\A$ to be the set \[ \cone(\A):=\left\{\sum_{i=0}^m\gamma_iv_i\,\text{ such that } 0\leq\gamma_i \right\}\subset \R^d. \] Notice that the conical hull is unbounded, as in particular it contains the rays $\R_{\geq0}\cdot v_i$ for {${0\leq i \leq m}$}. We are particularly interested in conical hulls of the following kind. Let $\A=\{v_0,\dots,v_d\}$ with $\Delta$ a lattice $d$-simplex. Then the {\emph{cone over}} $\Delta$ is \[ \cone\{(1,v_0),\dots,(1,v_d)\}\subset\R^{d+1} \, , \] and is denoted $\cone(\Delta)$. We next recall the fundamental parallelepiped, a distinguished subset of $\cone(\Delta)$. \begin{definition}For a lattice $d$-simplex $\Delta$ with vertices $v_0$ through $v_d$, the {\emph{fundamental parallelepiped}} $\Pi_\Delta$ is the set \[\Pi_\Delta\,:=\,\left\{\sum_{i=0}^{d}\gamma_i(1,v_i)\text{ such that }0\leq\gamma_i<1\right\}\subset\cone(\Delta).\] \end{definition} Interest in the fundamental parallelepiped $\Pi_\Delta$ arises mainly from the following well-known fact: every lattice point in $\cone(\Delta)$ can be written uniquely as a non-negative integer combination of the $(1,v_i)$'s and a lattice point in $\Pi_\Delta$. To see this, note that because any element $z$ of $\cone(\Delta)\cap\Z^{d+1}$ lies in $\cone(\Delta)$, it is a non-negative linear combination of the $(1,v_i)$'s, i.e., there exist non-negative real coefficients $g_i$ such that \[ z=\sum_{i=0}^{d}g_i(1,v_i)=\left(\sum_{i=0}^{d}\left\lfloor g_i\right\rfloor (1,v_i)\right) + \left(\sum_{i=0}^{d}\{g_i\} (1,v_i)\right) \] where $\{g_i\}$ means the fractional part of $g_i$. By setting $\gamma_i$ equal to $\{g_i\}$, we see that any point $z$ may be written as a non-negative integral combination of the $(1,v_i)$'s and an integer point in $\Pi_\Delta\cap\Z^{d+1}$. In particular, it is well-known that the set $\cone(\Delta)\cap\Z^{d+1}$ has a unique finite minimal additive generating set. \begin{definition}\label{def:hb} The unique minimal additive generating set of $\cone(\Delta)\cap\Z^{d+1}$ is called the {\emph{Hilbert basis}} $\HH$ of $\cone(\Delta)$. It consists of the $(1,v_i)$'s and some lattice points $h_1$ through $h_m$ in $\Pi_\Delta$ such that \[ \cone(\Delta)\cap\Z^{d+1}=\left\{\left(\sum_{i=0}^dr_i(1,v_i)\right)+\left(\sum_{j=1}^ms_i\,h_i\right)\text{ such that } r_i\,,\,s_j\in\Z_{\geq0}\right\} \, . \] \end{definition} The Hilbert basis consists of the cone generators $(1,v_i)$ together with the additively minimal elements $h_j$ of $\Pi_\Delta\cap\Z^{d+1}$. If the matrix whose columns are given by $(1,v_i)$ has determinant $\pm \v$, we say that the simplex $\Delta$ has {\emph{normalized volume}} $\v$. Since $\v$ is precisely the index of the sub-lattice generated by $(1,v_0)$ through $(1,v_d)$, we see that the normalized volume is equal to the number of lattice points in $\Pi_\Delta$. If the normalized volume of $\Delta$ is equal to one, then we call $\Delta$ a {\emph{unimodular}} simplex. The set of lattice points $\Z^{d+1}\cap\Pi_\Delta$ can be equipped with the following partial order, inherited from a well-known partial order on the lattice points in $\Z^{d+1}\cap\cone(\Delta)$. \begin{definition} The set $\Z^{d+1}\cap\Pi_\Delta$ is partially ordered by letting $\sigma\prec\mu$ if and only if $\mu-\sigma$ is an element of $\Z^{d+1}\cap\Pi_\Delta$. We call this poset the {\emph{fundamental parallelepiped poset}} $\PP(\Delta)$. \end{definition} Observe that the zero element of $\Z^{d+1}\cap\Pi_\Delta$ is below every other element of $\PP(\Delta)$, and that the minimal elements of $\PP(\Delta)\smallsetminus \{0\}$ are precisely the elements $h_1,\dots,h_m$ of the Hilbert basis of $\cone(\Delta)$. Our interest is in the case where the Hilbert basis contains all the elements of $\PP(\Delta)\smallsetminus \{0\}$, leading to the following definition. \begin{definition} If $\Delta$ is a simplex such that $P(\Delta)\smallsetminus \{0\}$ is an antichain, we call $\Delta$ an \emph{antichain simplex} and say $\Delta$ is \emph{antichain}. \end{definition} \begin{example} Recall that an empty simplex is one whose only lattice points are its vertices. It is known that any $2$-dimensional empty simplex is unimodularly equivalent to the convex hull of the origin and the standard basis vectors; this is easily verified to be antichain. Let $\Delta$ be a $3$-dimensional empty simplex, for which a classification of such simplices was given by G. K. White~\cite{WhiteSimplices}. Since $\Delta$ is empty, no lattice points of $\Pi_\Delta$ have $0$-th coordinate equal to one, and thus the only possible $0$-th coordinates of lattice points in $\Pi_\Delta$ are $2$ or $3$. However, sums of pairs of such lattice points have $0$-th coordinate equal to $4$, $5$, or $6$, and hence $\PP(\Delta)$ has no relations. Thus, every $3$-dimensional empty simplex is antichain. \end{example} When attempting to determine whether or not $\Delta$ is antichain, the first problem encountered is to enumerate the elements of $\Pi_\Delta\cap\Z^{d+1}$. As an initial attempt in this direction, let the matrix $A$ have columns given by $\{(1,v_i)\}_{0\leq i\leq d}$, where the $v_i$'s are the vertices of $\Delta$. Recall that the normalized volume $\v$, the number of elements of $\Pi_\Delta$, may be computed by $\v=|\det A|$. Recall also that the set $\Pi_\Delta$ is the image of $[0,1)^{d+1}$ under the linear transformation $A$, so that the preimage of a lattice point of $\Pi_\Delta$ must be a rational point of $[0,1)^{d+1}$ with denominator $\v$. We may therefore compute the set of points in $\Pi_\Delta\cap\Z^{d+1}$ by considering each element of the form \[ \left\{A\cdot\left(\frac{b_0}{\v},\cdots,\frac{b_d}{\v}\right)^T\text{ such that }0\leq b_i<\v\right\} \, , \] throwing out the ones which are not integer points. Unfortunately, this test set grows as $\v^{d+1}$, and there is no easy way to describe the lattice points among them. The software {\tt Normaliz}~\cite{Normaliz} gives a more efficient implementation based on the fact that (possibly after a lattice translation) the matrix $A$ has a representation $A=UH$ where $U$ is a unimodular matrix and $H$ is in Hermite normal form. Bruns et al.~\cite{bruns2012power} show that, for $\{c_{i,i}\}_{0\leq i\leq d}$ given by the diagonal entries of the matrix $H$, lattice points in \[[0,c_{0,0})\times\cdots\times[0,c_{d,d}) \] are representatives of the quotient classes (in $\Z^{d+1}$ modulo the $(1,v_i)$'s) of the elements of $\Pi_\Delta\cap\Z^{d+1}$. It is then sufficient to consider the image under $A$ of the elements $\left(A^{-1}\cdot x\right)\mod\Z^{d+1}$ for $x\in[0,c_{0,0})\times\cdots\times[0,c_{d,d})$. This modular arithmetic is implemented in a computer easily enough, but introduces number theory to any analysis of the poset $P(\Delta)$. \subsection{Lattice simplices with a unimodular facet and their posets}\label{sec:simplices} In this work, we study a restricted class of simplices in order to avoid both of the methods described above for determining $\Pi_\Delta\cap \Z^{d+1}$. \begin{definition} We say that a lattice $d$-simplex has a {\emph{unimodular facet}} if there exists a permutation $\pi$ in $\fS_{d+1}$ such that $\conv(\{v_{\pi_1},\dots,v_{\pi_d}\})$ is a unimodular lattice $(d-1)$-simplex. \end{definition} One way to represent these simplices is through their Hermite normal form, which is of the form \[ \left[ \begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0 & a_1 \\ 0 & 0 & 1 & \cdots & 0 & a_2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & a_{d-1} \\ 0 & 0 & 0 & \cdots & 0 & n \end{array}\right] \] where for $i=1,\ldots,d-1$ we have $0\leq a_iQ$. Thus, these are distinct classes of simplices. \end{remark} The following is a straightforward determinant calculation that also follows from the Hermite normal form given in Remark~\ref{rmk:hnf}. \begin{proposition}\label{prop:fpplatticepoints} The number of lattice points in $\Pi_\lambda$, which is equal to the normalized volume of $\Delta_\lambda$, is $\sum_{i=1}^d\lambda_i -1 = n-1$. \end{proposition} We can now describe the integer points in $\Pi_\lambda$ using only the entries of $\lambda$. \begin{proposition}\label{prop:fppparameter} For each integer $b$ with $0\leq b < n-1$, there is a unique lattice point $p(b)$ in $\Pi_\lambda$ given by \begin{equation}\label{eqn:p(b)} p(b) =\left(\left(\sum_{i=1}^d\left\lceil \frac{b\lambda_i}{n-1}\right\rceil\right)-b\;,\; \left\lceil \frac{b\lambda_1}{n-1}\right\rceil\;,\;\dots\;,\;\left\lceil \frac{b\lambda_d}{n-1}\right\rceil\right) \, . \end{equation} Every integer point in $\Pi_\lambda$ arises in this manner, and thus we identify the integer $b$ with the lattice point $p(b)$. \end{proposition} \begin{proof} For an element $\sum_{i=1}^d\gamma_i(1,e_i) + \gamma_{d+1}(1,\lambda) \in\Pi_\lambda\cap\Z^{d+1}$, we have \[ \left(\left(\sum_{i=1}^{d+1}\gamma_i\right)\;,\;(\gamma_1+\gamma_{d+1}\lambda_1)\;,\;\dots\;,\;(\gamma_d+\gamma_{d+1}\lambda_d)\right) \in \Z^{d+1}\, . \] Because of the condition that each $\gamma_i$ is strictly less than one, for each $i$ we have \[ \gamma_i=\lceil \gamma_{d+1}\lambda_i\rceil - \gamma_{d+1}\lambda_i \, , \] thus \begin{align*}&\left(\gamma_{d+1}+\sum_{i=1}^{d}\left(\lceil \gamma_{d+1}\lambda_i\rceil-\gamma_{d+1}\lambda_i\right),\lceil \gamma_{d+1}\lambda_1\rceil,\dots,\lceil \gamma_{d+1}\lambda_d\rceil\right)\\ =&\left(\gamma_{d+1}\left(1-\sum_{i=1}^{d}\lambda_i\right)+\sum_{i=1}^d\lceil \gamma_{d+1}\lambda_i\rceil,\lceil \gamma_{d+1}\lambda_1\rceil,\dots,\lceil \gamma_{d+1}\lambda_d\rceil\right).\end{align*} Observe that the first coordinate of this vector is an integer, hence \[ \gamma_{d+1}\left(1-\sum_{i=1}^{d}\lambda_i\right)=\gamma_{d+1}(1-n) \in \Z \, . \] It follows that $\gamma_{d+1}$ is a rational number of the form $b/(n-1)$, and every lattice point arises in this manner and is of the form \[ \left(\left(\sum_{i=1}^d\left\lceil \frac{b\lambda_i}{n-1}\right\rceil\right)-b\;,\; \left\lceil \frac{b\lambda_1}{n-1}\right\rceil\;,\;\dots\;,\;\left\lceil \frac{b\lambda_d}{n-1}\right\rceil\right) \, . \] Since there are $n-1$ lattice points in $\Pi_\lambda$ by Proposition~\ref{prop:fpplatticepoints}, there must be one unique lattice point for each $0\leq b0$ since $p(0)=0$. It follows that for all $1\leq t\leq d$, we have \[ \left\lceil \frac{i\lambda_t}{n-1}\right\rceil + \left\lceil \frac{\ell\lambda_t}{n-1}\right\rceil = \left\lceil \frac{j\lambda_t}{n-1}\right\rceil \, . \] Given this, we have that $p(i)_1+p(\ell)_1=p(j)_1$ reduces to $i+\ell=j$, forcing $\ell=j-i>0$, as desired. For the reverse direction, if $i=-2pt] (10) -- (bot1); \draw [thick, shorten <=-2pt, shorten >=-2pt] (10) -- (bot2); \draw [thick, shorten <=-2pt, shorten >=-2pt] (10) -- (bot3); \draw [thick, shorten <=-2pt, shorten >=-2pt] (10) -- (bot4); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top6) -- (bot2); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top6) -- (bot3); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top6) -- (bot4); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top7) -- (bot3); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top7) -- (bot4); \draw [thick, shorten <=-2pt, shorten >=-2pt] (top8) -- (bot4); \end{tikzpicture} \caption{The poset $P(6,2)$.} \end{center} \end{figure} \begin{proof} By Lemma~\ref{lem:ij}, we see that $i\prec j$ if and only if $i s_{t,j}>0$, \item $s_{t,i}=0$ and $s_{t,j}=s_{t,j-i}$, or \item $s_{t,j}=s_{t,i}>0$ and $s_{j-i}=0$. \end{enumerate} \end{theorem} \begin{proof} \emph{Forward implication:} Suppose that $i\prec j$ in $P(\lambda)$, and thus by Lemma~\ref{lem:scondition} the $s$-values satisfy~\eqref{eqn:seqn}. We consider five cases: \begin{itemize} \item $s_{t,i}=0$ \item $s_{t,i}>s_{t,j}=0$ \item $s_{t,i}=s_{t,j}>0$ \item $s_{t,i}>s_{t,j}>0$ \item $s_{t,j}>s_{t,i}>0$ \end{itemize} \emph{Case 1: $s_{t,i}=0$}. If $s_{t,i}=0$, then by~\eqref{eqn:seqn} we have that \[ \frac{s_{t,j-i}-s_{t,j}}{n-1} = \left\lceil \frac{s_{t,j-i}}{n-1}\right\rceil - \left\lceil \frac{s_{t,j}}{n-1}\right\rceil \, . \] Thus $\displaystyle \frac{s_{t,j-i}-s_{t,j}}{n-1}$ is equal to an integer, and the fact that $0\leq s_{t,\ell}s_{t,j}=0$}. In this case,~\eqref{eqn:seqn} implies \[ \frac{s_{t,i}+s_{t,j-i}}{n-1} = \left\lceil \frac{s_{t,i}}{n-1}\right\rceil + \left\lceil \frac{s_{t,j-i}}{n-1}\right\rceil \, . \] Thus $\displaystyle \frac{s_{t,i}+s_{t,j-i}}{n-1}$ is an integer, and again since $0\leq s_{t,\ell}s_{t,j}=0$. \emph{Case 3: $s_{t,i}=s_{t,j}>0$}. In this case,~\eqref{eqn:seqn} implies \[ \frac{s_{t,j-i}}{n-1} = \left\lceil \frac{s_{t,j-i}}{n-1}\right\rceil \, . \] This forces $s_{t,j-i}=0$, resulting in the third condition in the theorem statement. \emph{Case 4: $s_{t,i}>s_{t,j}>0$}. In this case,~\eqref{eqn:seqn} implies $\displaystyle \frac{s_{t,i}+s_{t,j-i}-s_{t,j}}{n-1}$ is equal to an integer, and the fact that every $0\leq s_{t,\ell}s_{t,i}-s_{t,j}>0$, we must have $\displaystyle \frac{s_{t,i}+s_{t,j-i}-s_{t,j}}{n-1}=1$, and also the right-hand side of~\eqref{eqn:seqn} is equal to $1$. Thus, the first condition in the theorem statement is possible if $i\prec j$. \emph{Case 5: $s_{t,j}>s_{t,i}>0$}. Following the same logic as in the previous case, we must have $\displaystyle \frac{s_{t,i}+s_{t,j-i}-s_{t,j}}{n-1}=0$ and thus $s_{t,j-i}\neq 0$. But then the right-hand side of~\eqref{eqn:seqn} is equal to $0$ while the right-hand side is equal to $1$, a contradiction. \emph{Reverse implication:} We verify that each of the three conditions listed in the theorem statement imply that~\eqref{eqn:seqn} is valid. First, by equation~\eqref{eqn:rtos} we have $\displaystyle \frac{s_{t,i}+s_{t,j-i}-s_{t,j}}{n-1}\in \Z$. Combining $n-1>s_{t,i}>s_{t,j}>0$ and the general bounds $0\leq s_{t,\ell}s_{t,i}-s_{t,j}>0$, we have $s_{t,j-i}=n-1-(s_{t,i}-s_{t,j})>0$, and thus \[ \left\lceil \frac{s_{t,i}}{n-1}\right\rceil + \left\lceil \frac{s_{t,j-i}}{n-1} \right\rceil - \left\lceil \frac{s_{t,j-i}}{n-1}\right\rceil =1\, . \] We conclude that equation~\eqref{eqn:seqn} holds. Second, if $s_{t,i}=0$ and $s_{t,j}=s_{t,j-i}$, then it is immediate that~\eqref{eqn:seqn} holds. Finally, if $s_{t,j}=s_{t,i}>0$ and $s_{j-i}=0$, then again it is immediate that~\eqref{eqn:seqn} holds. \end{proof} The following corollary illustrates a special case of Theorem~\ref{thm:order} that we will focus on in the remainder of this paper. \begin{corollary}\label{cor:order} Let $\lambda$ be a partition of $n$ where each coordinate is coprime to $n-1$, i.e., $\gcd(n-1,\lambda_t)=1$. Then $i\prec j$ in $P(\lambda)$ if and only if $s_{t,i}> s_{t,j}>0$ for every $t$. \end{corollary} \begin{proof} If $\gcd(n-1,\lambda_t)=1$, then $s_{t,i}\neq 0$ for all $i$. Thus, the second and third conditions in Theorem~\ref{thm:order} do not apply. \end{proof} \begin{remark} Ehrhart-theoretic properties of simplices $\Delta_\lambda$ that satisfy the relatively prime condition in Corollary~\ref{cor:order} have been previously studied by Hibi, Higashitani, and Li~\cite[Section 3]{HibiHigashitaniLi}. \end{remark} We can use Corollary~\ref{cor:order} to prove the following structural result regarding $P(\lambda)$ in the case where each part of $\lambda$ is coprime to $n-1$. \begin{theorem}\label{thm:duality} Let $\lambda$ be a partition of $n$ such that each $\lambda_t$ is coprime to $n-1$. Then $P(\lambda)$ is self-dual. \end{theorem} \begin{proof} We claim that $\phi:x\to n-1-x$ for $x\in [n-2]$ is an order-reversing poset isomorphism. It is clear that $\phi$ is a bijection. To see that $\phi$ is order-reversing, observe that by Corollary~\ref{cor:order}, we have that $i\prec j$ if and only if \begin{equation}\label{eq:dual1} s_{t,i}>s_{t,j} \text{ for all } t \, . \end{equation} Due to the fact that $\gcd(n-1,\lambda_t)=1$, we have that $s_{t,i}+s_{t,n-i-i}=n-1$ for all $i$ and $t$, and thus~\eqref{eq:dual1} holds if and only if \begin{equation} s_{t,n-1-j}>s_{t,n-1-i} \text{ for all } t \, . \end{equation} This final condition holds if and only if $n-1-j\prec n-1-i$, as desired. \end{proof} \subsection{Partitions with one distinct part}\label{sec:onedistinct} As an example of how these results can be applied, we consider the case where $\lambda=(x,x,\ldots,x)$ has $v$ occurrences of $x$. In this case, it is immediate that $x$ is coprime to $n-1=vx-1$. The following theorem shows that $P(\lambda)$ has a direct interpretation as a subposet of $\Z^2$. \begin{theorem}\label{thm:onepartposet} For $\lambda=(x,x,\ldots,x)$ with $v$ occurrences of $x$, we have that $P(\lambda)$ is isomorphic to the poset with elements \[ \left\{(r,p):0\leq rp'$ and $r'>r$. \end{theorem} \begin{proof} For $1\leq i\leq vx-2$, write \[ i=r_i v+p_i \] where $0\leq r_is_j$, which happens if and only if the following two conditions simultaneously occur: \begin{itemize} \item $p_i > p_j$ or $p_i=p_j$ with $r_i>r_j$ \item $r_j >r_i$ or $r_j = r_i$ with $p_j > p_i$ \end{itemize} The only way for both conditions to simultaneously occur is to have $p_i > p_j$ and $r_j >r_i$, and thus our proof is complete. \end{proof} The following corollary follows immediately. \begin{corollary}\label{cor:onedigitswap} The posets for $\lambda=(x,x,\ldots,x)$ where $x$ occurs $v$ times and $\lambda'=(v,v,\ldots,v)$ where $v$ occurs $x$ times are isomorphic. \end{corollary} Corollary~\ref{cor:onedigitswap} is interesting because the two lattice simplices corresponding to $\lambda$ and $\lambda'$ are in different dimensions. As an aside, we remark that the order on the lattice points within a rectangular grid given in Theorem~\ref{thm:onepartposet} corresponds to the reflexive closure of the direct product of two strict total orders. \begin{example} Figures~\ref{fig:46} and~\ref{fig:64} show the Hasse diagrams of the posets $P(4,4,4,4,4,4)$ and $P(6,6,6,6)$, respectively, embedded in $\Z^2$ as described in Theorem~\ref{thm:onepartposet}. This illustrates the isomorphism obtained by switching the roles of $x$ and $v$. \begin{figure}[ht] \centering \begin{minipage}{.5\textwidth} \centering \resizebox{0.5\textwidth}{!}{\includegraphics{posetfig46.eps}} \caption{$P(4,4,4,4,4,4)$.} \label{fig:46} \end{minipage} \begin{minipage}{.5\textwidth} \centering \resizebox{0.8\linewidth}{!}{\includegraphics{posetfig64.eps}} \caption{$P(6,6,6,6)$.} \label{fig:64} \end{minipage} \end{figure} \end{example} \section{Experimental results}\label{sec:experiments} \subsection{Exhaustive search of $\Delta_\lambda$ over all partitions of $n$} The results in Section~\ref{sec:fpp}, particularly Theorem~\ref{thm:order} and Corollary~\ref{cor:order}, provide explicit tools for studying the relations in $P(\lambda)$. Also, Corollary~\ref{cor:order} and Theorem~\ref{thm:duality} demonstrate that the condition that the parts of $\lambda$ be relatively prime to $\sum_i\lambda_i-1$ imposes additional structure on $P(\lambda)$, leading us to the following definition. \begin{definition} Given a partition $\lambda$ of $n$, if $\gcd(\lambda_i,n-1)=1$ for all $i$, we say $\lambda$ \emph{satisfies the relatively prime condition}. Let $\Ptn(n)$ denote the set of partitions of $n$, and let $\ptn(n):=|\Ptn(n)|$. Let $\Relprime(n)$ denote the set of partitions of $n$ that satisfy the relatively prime condition, and set $\relprime(n):=|\Relprime(n)|$. Finally, let $\Rpac(n)$ denote the subset of $\Relprime(n)$ for which $\Delta_\lambda$ is an antichain simplex, and set $\rpac(n):=|\Rpac(n)|$. \end{definition} Using {\tt SageMath}~\cite{SAGE} via {\tt CoCalc.com}~\cite{cocalc}, we computed $\ptn(n)$, $\relprime(n)$, and $\rpac(n)$ for all $1\leq n\leq 73$; the results are given in Table~\ref{table1}. Figure~\ref{fig:percentrelprime} plots $\rp(n)/\pn(n)$ for these values, and Figure~\ref{fig:percentantichainofrelprime} plots the ratio $\rpac(n)/\relprime(n)$. \begin{figure}[ht] \centering \includegraphics[width=0.7\textwidth]{percent_relatively_prime.eps} \caption{The ratio $\rp(n)/\pn(n)$ for $1\leq n\leq 73$.} \label{fig:percentrelprime} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=0.7\textwidth]{percent_antichain_of_relativelyprime.eps} \caption{The ratio $\rpac(n)/\relprime(n)$ for $1\leq n\leq 73$.} \label{fig:percentantichainofrelprime} \end{figure} What follows are some observations regarding this experimental data. \begin{enumerate} \item Figure~\ref{fig:percentantichainofrelprime} shows that regardless of the total value of $\relprime(n)$, $\rpac(n)/\relprime(n)$ is generally above $0.8$ and as $n$ grows it is clustering between $0.85$ and $0.95$. Thus, these experiments suggest that when $\lambda$ satisfies the relatively prime condition, it is likely that $\Delta_\lambda$ is antichain. \item Figure~\ref{fig:percentrelprime} shows that when $n-1$ is not prime or the square of a prime, the ratio $\rp(n)/\pn(n)$ appears to be small, and thus our consideration of the relatively prime condition does not broadly apply to partitions in this case. However, it is immediate that when $n-1$ is prime, every partition of $n$ except for $1+(n-1)$ satisfies the relatively prime condition, and thus $\rpac(n)/\relprime(n)=\rpac(n)/(\pn(n)-1)$. Thus, it appears that one likely source of antichain simplices are those $\Delta_\lambda$ for which $n-1$ is prime. \item In Figure~\ref{fig:percentantichainofrelprime}, the values of $\rpac(n)/\relprime(n)$ for $n\geq 13$ that lie on the upper hull of the data plot arise from $n$ in $\{13, 19, 31, 43, 61, 67, 73\}$. These are all prime numbers, and an OEIS~\cite{oeis} search finds that these values arise in three known sequences, including the sequence \seqnum{A040047} of those primes $p$ such that $x^3=6$ has no solution mod $p$. \item Again in Figure~\ref{fig:percentantichainofrelprime}, the values of $\rpac(n)/\relprime(n)$ for $70\geq n\geq 20$ that lie on the lower hull of the data plot arise from $n$ in $\{20, 26, 32, 38, 44, 50, 62, 68\}$. These values are all of the form $6k+2$, though whether due to coincidence or mathematics it is not clear. \item When $n-1$ is a superabundant \seqnum{A004394} or highly composite \seqnum{A002182} number, one might expect to see particularly low numbers of relatively prime antichain simplices, which is supported by the data given in Table~\ref{table1}. \end{enumerate} At this time, the authors do not have an explanation for why the ratio $\rpac(n)/\relprime(n)$ appears to be clustering as $n$ grows, leading to the following problems. \begin{problem} Determine if there is a limiting value to which the sequence $\rpac(n)/\relprime(n)$ converges as $n$ increases. Alternatively, determine if there are any connections between a liminf or limsup value for $\rpac(n)/\relprime(n)$ and the values of $n$ corresponding to subsequences achieving those values, as hinted at in the observations above. \end{problem} \begin{table} \centering \begin{tabular}{|l|l|l|l|} \hline $n$ & rpac($n$) & relprime($n$) & part($n$) \\ \hline \hline $1$ & $1$ & $1$ & $1$ \\ $2$ & $2$ & $2$ & $2$ \\ $3$ & $2$ & $2$ & $3$ \\ $4$ & $3$ & $4$ & $5$ \\ $5$ & $3$ & $3$ & $7$ \\ $6$ & $7$ & $10$ & $11$ \\ $7$ & $3$ & $3$ & $15$ \\ $8$ & $15$ & $21$ & $22$ \\ $9$ & $7$ & $8$ & $30$ \\ $10$ & $17$ & $22$ & $42$ \\ $11$ & $8$ & $8$ & $56$ \\ $12$ & $58$ & $76$ & $77$ \\ $13$ & $7$ & $7$ & $101$ \\ $14$ & $103$ & $134$ & $135$ \\ $15$ & $18$ & $21$ & $176$ \\ $16$ & $45$ & $56$ & $231$ \\ $17$ & $33$ & $38$ & $297$ \\ $18$ & $316$ & $384$ & $385$ \\ $19$ & $15$ & $16$ & $490$ \\ $20$ & $513$ & $626$ & $627$ \\ $21$ & $36$ & $41$ & $792$ \\ $22$ & $180$ & $215$ & $1002$ \\ $23$ & $78$ & $89$ & $1255$ \\ $24$ & $1317$ & $1574$ & $1575$ \\ $25$ & $31$ & $34$ & $1958$ \\ $26$ & $1169$ & $1414$ & $2436$ \\ $27$ & $148$ & $170$ & $3010$ \\ $28$ & $750$ & $874$ & $3718$ \\ $29$ & $143$ & $162$ & $4565$ \\ $30$ & $4779$ & $5603$ & $5604$ \\ $31$ & $26$ & $28$ & $6842$ \\ $32$ & $7050$ & $8348$ & $8349$ \\ $33$ & $392$ & $448$ & $10143$ \\ $34$ & $1675$ & $1951$ & $12310$ \\ $35$ & $478$ & $539$ & $14883$ \\ $36$ & $4850$ & $5625$ & $17977$ \\ $37$ & $115$ & $126$ & $21637$ \\ $38$ & $22109$ & $26014$ & $26015$ \\ $39$ & $816$ & $918$ & $31185$ \\ $40$ & $4410$ & $5047$ & $37338$ \\ \hline \end{tabular} \quad \begin{tabular}{|l|l|l|l|} \hline $n$ & rpac($n$) & relprime($n$) & part($n$) \\ \hline \hline $41$ & $433$ & $481$ & $44583$ \\ $42$ & $45819$ & $53173$ & $53174$ \\ $43$ & $104$ & $112$ & $63261$ \\ $44$ & $64731$ & $75174$ & $75175$ \\ $45$ & $1362$ & $1522$ & $89134$ \\ $46$ & $4192$ & $4747$ & $105558$ \\ $47$ & $2202$ & $2468$ & $124754$ \\ $48$ & $129242$ & $147272$ & $147273$ \\ $49$ & $365$ & $399$ & $173525$ \\ $50$ & $106948$ & $123165$ & $204226$ \\ $51$ & $1233$ & $1362$ & $239943$ \\ $52$ & $24641$ & $27874$ & $281589$ \\ $53$ & $3597$ & $3986$ & $329931$ \\ $54$ & $339300$ & $386154$ & $386155$ \\ $55$ & $623$ & $679$ & $451276$ \\ $56$ & $128590$ & $145176$ & $526823$ \\ $57$ & $3426$ & $3781$ & $614154$ \\ $58$ & $54230$ & $60927$ & $715220$ \\ $59$ & $8575$ & $9496$ & $831820$ \\ $60$ & $864231$ & $966466$ & $966467$ \\ $61$ & $302$ & $324$ & $1121505$ \\ $62$ & $1146930$ & $1300155$ & $1300156$ \\ $63$ & $13151$ & $14458$ & $1505499$ \\ $64$ & $55541$ & $61850$ & $1741630$ \\ $65$ & $16496$ & $18200$ & $2012558$ \\ $66$ & $522255$ & $586074$ & $2323520$ \\ $67$ & $1012$ & $1091$ & $2679689$ \\ $68$ & $2761384$ & $3087734$ & $3087735$ \\ $69$ & $20580$ & $22503$ & $3554345$ \\ $70$ & $234794$ & $261034$ & $4087968$ \\ $71$ & $3040$ & $3287$ & $4697205$ \\ $72$ & $4875893$ & $5392782$ & $5392783$ \\ $73$ & $2715$ & $2931$ & $6185689$ \\ & & & \\ & & & \\ & & & \\ & & & \\ & & & \\ & & & \\ & & & \\ \hline \end{tabular} \caption{Experimental data} \label{table1} \end{table} \subsection{Random sampling of simplices with one non-trivial column in Hermite normal form} It is worthwhile to compare the results for our restricted $\Delta_\lambda$ simplices to arbitrary simplices with Hermite normal form given by \[ \left[ \begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0 & a_1 \\ 0 & 0 & 1 & \cdots & 0 & a_2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & a_{d-1} \\ 0 & 0 & 0 & \cdots & 0 & n \end{array}\right] \] where for $i=1,\ldots,d-1$ we have $0\leq a_i