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The Fundamental Lemma of (P, w)-partitions.tex @@ -0,0 +1,500 @@ +\documentclass[11pt]{amsart} + +% Include Packages +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{amsxtra} +\usepackage{caption} +\usepackage{epsfig,color} +\usepackage{enumerate} +\usepackage[dvipsnames]{xcolor} +\usepackage{fullpage} +\usepackage{hyperref} +\usepackage{mathtools} +\usepackage{vmargin} +\usepackage{MnSymbol} +% Tikz package and libraries +\usepackage{tikz} +\usepackage{tikz-cd} +%\usepackage{tikzducks} +\usetikzlibrary{arrows, backgrounds, chains, decorations.pathreplacing, patterns, shapes, calc, positioning} + +% Theorem Commands +\theoremstyle{plain} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem*{claim}{Claim} +\newtheorem{conjecture}{Conjecture} + +\theoremstyle{definition} +\newtheorem{definition}{Definition} +\newtheorem*{question}{Question} +\newtheorem*{notation}{Notation} +\newtheorem{exercise}{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}{Remark}[section] +\newtheorem{example}{Example}[section] +\newtheorem{examples}{Examples} + +\newcommand{\RR}{\mathbb{R}} +\newcommand{\LL}{\mathcal{L}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\NN}{\mathbb{N}} + +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\Des}{Des} +\DeclareMathOperator{\maj}{maj} + + +\newcommand{\binomsq}[2]{\left[\begin{matrix} #1 \\ #2 \end{matrix}\right]} + +\numberwithin{equation}{section} + +\setmarginsrb{33mm}{30mm}{33mm}{40mm}% + {0mm}{10mm}{0mm}{10mm} + + +\begin{document} + +\begin{center} + \huge{Algebraic combinatorics} +\end{center} + +\bigskip + +\begin{center} + \Large{Lecture 17: The Fundamental Lemma of $(P, w)$-partitions (20/11/2025)} +\end{center} + +\bigskip + +\begin{center} + \Large{Michele D'Adderio} +\end{center} +\smallskip +\begin{center} + \large{(notes by Gabriel Antonio Videtta)} +\end{center} +\bigskip +\begin{center} + \textcolor{red}{ + WARNING: any information contained in these notes can be wrong!}\\ + \textcolor{red}{READER DISCRETION IS ADVISED} +\end{center} + +\tableofcontents + +\section{Labelled posets and \texorpdfstring{$(P, w)$}{(P, w)}-partitions} + +We're looking to find a suitable statistic to associate to $\Omega(P, n)$, +the order-reversing\footnote{ + This is merely a convention. The theory concerning order-preserving maps is + studied using essentially the same arguments, with appropriate adjustments. +} functions from a poset $P$ to a chain of length $n$ (e.g., $\{1 \leq 2 \leq \cdots \leq n\}$). \medskip + +As a first approach, we will define a generating function tracking \textit{every} aspect of such functions, +and we will restrict ourselves to a more particular statistic in the future lectures. + +In order to do so, we will assume we have a fixed \textit{labelling} $w$ for our poset $P$: + +\begin{definition} + Let $P$ be a poset with $n$ elements. A \textit{labelling of $P$} is a bijection $w : P \to [n]$. +\end{definition} + +\begin{figure}[h] + \centering + \begin{tikzpicture} + \begin{scope}[every node/.style={circle,draw}] + \node (A) at (0,0) [label=left:$t_1$] {1}; + \node (B) at (1.5,0) [label=right:$t_2$] {3}; + \node (C) at (0,1.5) [label=left:$t_3$] {2}; + \node (D) at (1.5,1.5) [label=right:$t_4$] {4}; + \node (E) at (0,3) [label=left:$t_5$] {5}; + \end{scope} + + \draw[->] (A) -- (C); + \draw[->] (B) -- (C); + \draw[->] (B) -- (D); + \draw[->] (C) -- (E); + + \end{tikzpicture} + \caption{A \textit{labelled} poset $\{t_1, t_2, t_3, t_4, t_5\}$, with + $t_1$, $t_2 \leq t_3 \leq t_5$ and $t_2 \leq t_4$. The labels are represented within + the circled nodes.} + \label{fig:example_labelled_poset} +\end{figure} + +We are now ready to state the definition for a $(P, w)$-partition, which allows us to be more flexible on +the choice of the ``weaknesses'' of our order-reversing maps: + +\begin{definition} + A $(P, w)$-partition is a map $\sigma : P \to \NN$ with the following two properties: + + \begin{enumerate}[(i.)] + \item \textit{it's (weakly) order-reversing:} if $s <_P t$, then $\sigma(s) \geq \sigma(t)$; + \item \textit{it respects the strictness according to the labelling:} if $s <_P t$ and + $w(s) > w(t)$, then $\sigma(s) > \sigma(t)$. + \end{enumerate} +\end{definition} + +\begin{notation} + We will denote the set of all $(P, w)$-partitions of $P$ with $A(P, w)$. +\end{notation} + +\begin{example} + For Figure \ref{fig:example_labelled_poset}, the resulting conditions for a map $\sigma$ to be + a $(P, w)$-partition are: + \begin{itemize} + \item being (weakly) order-reversing; + \item $\sigma(t_2) > \sigma(t_3)$, since $t_2 < t_3$ and $w(t_2) = 3 > 2 = w(t_3)$. + \end{itemize} +\end{example} + +\begin{remark} + If we choose a labelling $w$ such that, whenever $s <_P t$, $w(s) < w(t)$, a $(P, w)$-partition is + simply an order-reversing map -- it must satisfy only (i.), since (ii.) is vacuously true. + + Likewise, if we choose $w$ such that $w(s) > w(t)$, then the map must be \textit{strictly} order-reversing. +\end{remark} + +These observations lead us to the following definitions: + +\begin{definition} + We say a labelling $w$ is \textit{natural} if, whenever $s <_P t$, $w(s) < w(t)$. We say $w$ is \textit{dually natural} if + $w(s) > w(t)$ whenever $s <_P t$. \smallskip +\end{definition} + +For (dual) natural labellings, the conditions for a map to be $(P, w)$-partition do not depend on $w$. Therefore they are simply +called (strict) $P$-partitions. \smallskip + +We are now ready to define a statistics on the $(P, w)$-partitions: + +\begin{definition} + Let $P = \{t_1, t_2, \ldots, t_n\}$ be a poset with labelling $w$. We then define the \textit{$(P, w)$-partition enumerator} as: + \[ + F_{P, w} \coloneq F_{P, w}(x_1, \ldots, x_n) \coloneq \sum_{\sigma \in A(P, w)} x_1^{\sigma(t_1)} \cdots x_n^{\sigma(t_n)}. + \] +\end{definition} + +\begin{remark} + Given a natural labelling $w$, the generating function $F_{P, w}$ enumerates all the order-reversing maps from the $\Omega(P, n)$'s. To restrict + to a certain $n$, we simply ignore any monomials that contain a variable with degree greater than $n$. +\end{remark} + +The following examples provide insight into why the $(P, w)$-partition enumerator, in a certain sense, generalizes the partitions we've studied on the integers. + +\begin{example} \label{ex:chain_n} + Let $P$ be a naturally labelled chain $t_1 < t_2 < \cdots < t_n$. Recall that in this case a $P$-partition is the same as an order-reversing map. Thus: + \[ + F_P = \sum_{a_1 \geq a_2 \geq \cdots a_n \geq 0} x_1^{a_1} \cdots x_n^{a_n} = \frac{1}{(1-x_1) (1-x_1 x_2) \cdots (1 - x_1 x_2 \cdots x_n)}. + \] +\end{example} + +\begin{example} + Let $P$ be a dually naturally labelled chain $t_1 < t_2 < \cdots < t_n$. In this case a $P$-partition is the same as a \textit{strict} order-reversing map. + Therefore: + \[ + F_P = \sum_{a_1 > a_2 > \cdots a_n > 0} x_1^{a_1} \cdots x_n^{a_n} = \frac{x_1^{n-1} x_2^{n-2} \cdots x_{n-1}}{(1-x_1) (1-x_1 x_2) \cdots (1 - x_1 x_2 \cdots x_n)}. + \] +\end{example} + +\begin{example} \label{ex:antichain_n} + Let $P$ be an antichain (i.e., no comparisons are possible) of size $n$. It is a simple \textbf{exercise} to check that + $(P, w)$-partitions do not depend on $w$ and are simply maps from $P$ to $\NN$. Hence: + \[ F_P = \frac{1}{(1-x_1) \cdots (1-x_n)}. \] + + \begin{remark} + $F_P$ enumerates, in a certain sense, the possible partitions on $[n]$. Choosing $x_i = x^i$, we get the \textit{partition enumerator} series. + \end{remark} +\end{example} + +\begin{notation} + Let $P$ be a poset of size $n$ with labelling $w$. + We will employ the notation $\LL(P, w)$ to denote the set of permutations $\pi = \pi_1 \cdots \pi_n \in S_n$ such that the map $\sigma : P \to [n]$ defined + by $\sigma(w^{-1}(\pi_i)) = i$ is a linear extension (i.e., an order-preserving map) of $P$. +\end{notation} + +\begin{remark} + Let $P = [n]$ be a poset. There is a natural labelling on $P$, namely the identity map $\id \in S_n$. This simplifies + the definition of $\LL(P, w)$: + + \vspace{0.1in} + + \begin{quote} + $\LL(P, w)$ is the set of permutations $\pi = \pi_1 \cdots \pi_n \in S_n$ such that the map $\sigma : P \to [n]$ defined + by $\sigma(\pi_i) = i$ is a linear extension (i.e., an order-preserving map) of $P$. + \end{quote} + + \vspace{0.1in} + + In other words: + + \vspace{0.1in} + + \begin{quote} + $\LL(P, w)$ is the set of permutation $\pi \in S_n$ such that if $i \leq_P j$, then $i$ comes before $j$ in + the one-line notation of $\pi$. + \end{quote} +\end{remark} + +\begin{example} + \begin{figure}[h] + \centering + \begin{tikzpicture} + \begin{scope}[every node/.style={circle,draw}] + \node (A) at (0,0) [label=left:$t_1$] {1}; + \node (B) at (1.5,0) [label=right:$t_2$] {3}; + \node (C) at (0,1.5) [label=left:$t_3$] {2}; + \node (D) at (1.5,1.5) [label=right:$t_4$] {4}; + \end{scope} + + \draw[->] (A) -- (C); + \draw[->] (B) -- (C); + \draw[->] (B) -- (D); + + \end{tikzpicture} + \caption{A \textit{labelled} poset $\{t_1, t_2, t_3, t_4\}$, with + $t_1$, $t_2 \leq t_3$ and $t_2 \leq t_4$. The labels are represented within + the circled nodes.} + \label{fig:example_Lpw} + \end{figure} + + Let's compute $\LL(P, w)$ for the labelled poset from Figure \ref{fig:example_Lpw}. + $1$ and $3$ must come before $2$; $3$ must come before $4$ as well. Therefore: + \[ \LL(P, w) = \{ 1324, 1342, 3124, 3142, 3412 \}. \] +\end{example} + +\section{\texorpdfstring{$\pi$}{π}-compatibility and the Fundamental Lemma of \texorpdfstring{$(P, w)$}{(P, w)}-partitions} + +\begin{definition} + Let $\pi$ be a permutation of $n$ elements. A function $f : [n] \to \NN$ is said to be \textit{$\pi$-compatible} if: + + \begin{itemize} + \item \textit{it weakly decreases along the one-line notation of $\pi$:} $f(\pi_1) \geq f(\pi_2) \geq \cdots \geq f(\pi_n)$. + \item \textit{strictness is forced upon a descent:} $f(\pi_i) > f(\pi_{i+1})$ if $i$ is a descent (i.e., if $\pi_i > \pi_{i+1}$). + \end{itemize} +\end{definition} + +\begin{lemma} + Let $f$ be a function from $[n]$ to $\NN$. Then, there exists a unique permutation $\pi \in S_n$ such that + $f$ is $\pi$-compatible. \label{lem:unique_pi_permutation} +\end{lemma} + +\begin{proof} + Consider the unique (\textbf{exercise}) ordered set partition $(B_1, B_2, \ldots, B_k)$ such that + $f \big|_{B_i}$ is constant and $f \big|_{B_1} > f \big|_{B_2} > \cdots > f \big|_{B_k}$. (continue) + \begin{example} + Let $f : [5] \to \NN$ be such that: + + \begin{itemize} + \item $f(2) = f(3) = 1$; + \item $f(1) = f(4) = 6$; + \item $f(5) = 8$. + \end{itemize} + + Thus, $B_1 = \{5\}$, $B_2 = \{1, 4\}$ and $B_3 = \{2, 3\}$, since $8 > 6 > 1$. + \end{example} + + We are now ready to construct our permutation $\pi$ by specifying its one-line notation. + First, we order the elements within each block $B_i$ in increasing order: + \[ + B_i = \{ b_{i,1} \leq b_{i,2} \leq \cdots \leq b_{i, j_i} \}. + \] + We then define $\pi$ by concatenating these ordered blocks: + \[ + \pi = b_{1,1} \cdots b_{1, j_1} b_{2,1} \cdots b_{2, j_2} \cdots b_{k, j_k}. + \] + One can verify that $f$ is indeed $\pi$-compatible (\textbf{exercise}). The uniqueness of the permutation + $\pi$ follows straightforwardly from the uniqueness of the set partition $(B_1, B_2, \ldots, B_k)$. +\end{proof} + +\begin{definition} + Let $\sigma$ be a map from $P$ to $\NN$, where $(P, w)$ is a labelled poset of size $n$. We then define + the map $\sigma'$ from $[n]$ to $\NN$ to be the re-indexed version of $\sigma$ via $w$, where the label $w(p) \in [n]$ + is used instead of the poset element $p \in P$. In other words, $\sigma'$ is such that: + \[ + \sigma'(i) = \sigma(w^{-1}(i)). + \] +\end{definition} + +\begin{notation} + Let $(P, w)$ be a labelled poset of size $n$, and let $\pi \in S_n$ be a permutation of $n$ elements. We will employ the notation $S_\pi$ + to denote the set of all maps $\sigma$ from $P$ to $\NN$ such that their re-indexed versions $\sigma'$ are $\pi$-compatible. +\end{notation} + +\begin{theorem}[Fundamental Lemma of $(P, w)$-partitions] + The set of $(P, w)$-partitions $A(P, w)$ is exactly the disjoint union of the $S_\pi$'s, where $\pi$ varies over $\LL(P, w)$: + \[ + A(P, w) = \bigsqcup_{\pi \in \LL(P, w)} S_\pi. + \] \label{thm:fundamental_lemma} +\end{theorem} + +\begin{proof} + The disjointness of the right hand side follows straightforwardly from Lemma \ref{lem:unique_pi_permutation}. + We now prove both inclusions. + + \begin{itemize} + \item[($\subseteq$)] Let $\sigma \in A(P, w)$. Lemma \ref{lem:unique_pi_permutation} tells us there exists a unique + $\pi$ such that $\sigma \in S_\pi$. It remains only to verify that $\pi$ is an element of $\LL(P, w)$. + + Let $\pi = \pi_1 \cdots \pi_n$. Suppose $i < j$, $w(s) = \pi_i$ and $w(t) = \pi_j$. We need to show that + $s \not> t$. + + Since $\sigma \in S_\pi$, then: + \[ \sigma(s) = \sigma'(\pi_i) \geq \sigma'(\pi_j) = \sigma(t). \] + If we had $\sigma(s) > \sigma(t)$, then $s \not> t$, since $\sigma$ is order-reversing. Suppose then + $\sigma(s) = \sigma(t)$, i.e., $\sigma'(\pi_i) = \sigma'(\pi_j)$. Therefore we have: + \[ + \pi_i < \pi_{i+1} < \cdots < \pi_j, + \] + since there cannot be descents (otherwise we'd have $\sigma(s) > \sigma(t)$, \textbf{exercise}). Thus + $w(s) = \pi_i < \pi_j = w(t)$: if we had $s > t$, we would have $\sigma(t) > \sigma(s)$, which contradicts + our supposition. It follows that $s$ cannot be greater than $t$, and then that $A(P, w) \subseteq \bigsqcup_{\pi \in \LL(P, w)} S_\pi$. + + \item[($\supseteq$)] Let $\sigma$ be a map from $P$ to $\NN$ such that $\sigma'$ is $\pi$-compatible, with $\pi \in \LL(P, w)$. We need to show + that $\sigma$ is an element of $A(P, w)$. We will do so by showing that $\sigma$ satisfies the two conditions for being a $(P, w)$-partition. + + Let $s$, $t$ be elements of $P$ such that $s < t$. Let $\pi_i = w(s)$ and $\pi_j = w(t)$. Since + $\pi$ is an element of $\LL(P, w)$, then $\pi_i$ must come before $\pi_j$ in the one-line notation of $\pi$, i.e., $i < j$. Since $\sigma'$ is + $\pi$-compatible, we have $\sigma(s) = \sigma'(\pi_i) \geq \sigma'(\pi_j) = \sigma(t)$. + + Suppose now that $w(s) > w(t)$. Then, $\pi_i$ is strictly bigger than $\pi_j$, meaning there exists a descent $k$ with $i \leq k < j$. Therefore: + \[ + \sigma(s) = \sigma'(\pi_i) \geq \sigma'(\pi_{i+1}) \geq \cdots \geq \sigma'(\pi_k) > \sigma'(\pi_{k+1}) \geq \cdots \geq \sigma'(\pi_j) = \sigma(t). + \] + This concludes our proof. + \end{itemize} +\end{proof} + +The usefulness of the \nameref{thm:fundamental_lemma} comes from the fact that the generating function $F_{P, w}$ is easier to describe on $S_\pi$, as shown in the following result. + +\begin{definition} + Let $\pi \in S_n$ be a permutation of $n$ elements, and let $P = \{t_1, \ldots, t_n\}$ be a poset with size $n$ and labelling $w$. We then define a new + permutation $\pi' \in S_n$, such that $\pi'$ gives the index of the element in $P$ corresponding to the value in $\pi$. Specifically, $\pi'$ is such that: + \[ + \pi_i = w(t_j) \implies \pi_i' = j. + \] +\end{definition} + +\begin{lemma} + Let $\pi \in S_n$ be a permutation of $n$ elements, and let $P = \{t_1, t_2, \ldots, t_n\}$ be a poset with size $n$ and labelling $w$. Then: + \[ + F_{P, w, \pi} \coloneq \sum_{\sigma \in S_\pi} x_1^{\sigma(t_1)} \cdots x_n^{\sigma(t_n)} = \frac{\prod_{j \in \Des(\pi)} x_{\pi_1'} x_{\pi_2'} \cdots x_{\pi_j'}}{\prod_{i=1}^n (1 - x_{\pi_1'} \cdots x_{\pi_i'})}. + \] \label{lem:F_pi} +\end{lemma} + +\begin{proof} + \underline{This part is happily left to the professor to fill.} +\end{proof} + +As an immediate application of Lemma \ref{lem:F_pi} and the \nameref{thm:fundamental_lemma}, we get the following theorem (\textbf{exercise}): + +\begin{theorem} + Let $P = \{t_1, t_2, \ldots, t_n\}$ be a poset with size $n$ and labelling $w$. Then: + \[ + F_{P, w} = \sum_{\pi \in \LL(P, w)} \frac{\prod_{j \in \Des(\pi)} x_{\pi_1'} x_{\pi_2'} \cdots x_{\pi_j'}}{\prod_{i=1}^n (1 - x_{\pi_1'} \cdots x_{\pi_i'})}. + \] \label{thm:F_pw} +\end{theorem} + +\section{Applications of the Fundamental Lemma} + +Let's specialise the series from Theorem \ref{thm:F_pw} and get its $q$-analogue. We will denote +it with $F_{P, w}(q) := F_{P, w}(q, q, \ldots, q)$. + +\begin{example} + Let $P$ be a chain of size $n$ with a natural labelling. Thus $\LL(P, w)$ consists of only the identity $\id \in S_n$, with $\maj(\id) = 0$. + Therefore: \label{ex:Fq_chain} + \[ + F_P(q) = \frac{q^{\maj(\pi)}}{(1-q)(1-q^2) \cdots (1-q^n)} = \frac{1}{(1-q)(1-q^2) \cdots (1-q^n)}, + \] + which is consistent with what we'd already seen in Example \ref{ex:chain_n} +\end{example} + +The \nameref{thm:fundamental_lemma} and Theorem \ref{thm:F_pw} allow us to provide alternative +proofs for some of the theorems established in previous lectures. + +\begin{example} + Let $P$ be an antichain of size $n$. Thus $\LL(P, w) = S_n$. Using the results from Example \ref{ex:antichain_n}, we get: + \[ + \frac{1}{(1-q)^n} = F_P(q) = \frac{\sum_{\pi \in S_n} q^{\maj(\pi)}}{(1-q)(1-q^2) \cdots (1-q^n)}, + \] + from which the MacMahon theorem follows: + \[ + \sum_{\pi \in S_n} q^{\maj(\pi)} = \prod_{i=1}^n \frac{1-q^i}{1-q} = \prod_{i=1}^n [i]_q = [n]_q ! \, . + \] +\end{example} + +\begin{example} + Let $P = C_n + C_k$ be a disjoint union of two chains (see Figure \ref{fig:example_disjoint_union_two_chains}). + + \begin{figure}[h] + \centering + \begin{tikzpicture} + % C_n + \node[circle, fill, inner sep=1.5pt] (A) at (0,1.5) {}; + \node at (0, 1.1) {$1$}; + + \node[circle, fill, inner sep=1.5pt] (B) at (1.5,1.5) {}; + \node at (1.5, 1.1) {$2$}; + + \node[circle, fill, inner sep=1.5pt] (C) at (3,1.5) {}; + \node at (3, 1.1) {$n-1$}; + + \node[circle, fill, inner sep=1.5pt] (D) at (4.5,1.5) {}; + \node at (4.5, 1.1) {$n$}; + + % C_k + + \node[circle, fill, inner sep=1.5pt] (A1) at (0,0) {}; + \node at (0, -0.4) {$n+1$}; + + \node[circle, fill, inner sep=1.5pt] (B1) at (1.5,0) {}; + \node at (1.5, -0.4) {$n+2$}; + + \node[circle, fill, inner sep=1.5pt] (C1) at (3,0) {}; + \node at (3, -0.4) {$n+k-1$}; + + \node[circle, fill, inner sep=1.5pt] (D1) at (4.5,0) {}; + \node at (4.5, -0.4) {$n+k$}; + + \draw[->] (A) -- (B); + \draw[loosely dotted] (B) -- (C); + \draw[->] (C) -- (D); + + \draw[->] (A1) -- (B1); + \draw[loosely dotted] (B1) -- (C1); + \draw[->] (C1) -- (D1); + + \end{tikzpicture} + \caption{The \textit{labelled} disjoint union of two chains, one of length $n$ and the other of length $k$.} + \label{fig:example_disjoint_union_two_chains} + \end{figure} + + One can easily verify (\textbf{exercise}) that: + \[ + F_P(q) = F_{C_k}(q) \cdot F_{C_n}(q). + \] + Recall from Example \ref{ex:Fq_chain} that the following identity holds: + \[ + F_{C_n}(q) = \prod_{i=1}^n \frac{1}{1-q^i}, \qquad F_{C_k}(q) = \prod_{j=1}^k \frac{1}{1-q^j}. + \] + We can identify $\LL(P)$ with words with $n$ $0$'s and $k$ $1$'s, with the zeros representing elements from $C_n$ and the + ones representing elements from $C_k$ (\textbf{exercise}). Thus, by applying Theorem \ref{thm:F_pw} we get: + \[ + \left( \prod_{i=1}^n \frac{1}{1-q^i} \right) \left( \prod_{j=1}^k \frac{1}{1-q^j} \right) = \frac{\sum_{\omega \in R(0^n1^k)} q^{\maj(\omega)}}{\prod_1^{n+k} (1-q^i)}, + \] + from which the following identity follows again (\textbf{exercise}): + \[ + \binomsq{n+k}{k}_q = \sum_{\omega \in R(0^n1^k)} q^{\maj(\omega)}. + \] +\end{example} + +\end{document} + +