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