Some results on counting linearizations of posets

TL;DR

This paper investigates counting linearizations and sign-imbalances of posets, generalizes known results, and explores lexicographic sums of finite posets, providing new formulas and theoretical insights.

Contribution

It introduces new results on counting linearizations of posets, generalizes Stanley's sign-imbalance result, and analyzes lexicographic sums of posets with explicit formulas.

Findings

Derived a number-theoretic formula for counting certain isotone maps.

Generalized Stanley's sign-imbalance result for posets with uniform chain parity.

Provided methods to compute linearization counts and sign-imbalances for lexicographic sums.

Abstract

In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the number of maps $\varphi:|S|\to\{1,\dots,n\}$ which are both isotone with respect to the ordering $\preccurlyeq,$ and have the property that $\varphi(x)\neq \varphi(y)$ whenever $\{x,y\}\in E.$ We prove a number-theoretic result about this function, and use it in section 7 to recover a ring-theoretic identity of G. P. Hochschild. In section 2 we generalize a result of R. Stanley on the sign-imbalance of posets in which the lengths of all maximal chains have the same parity. In sections 3-6 we study the linearization-count and sign-imbalance of a lexicographic sum of $n$ finite posets $P_i$ $(1\leq i\leq n)$ over an $n$-element poset $P_0.$ We note how to compute these values from the corresponding counts for the given posets $P_i,$ and for a lexicographic sum over $P_0$ of chains of lengths $\mathrm{card}(P_i).$ This makes the behavior of lexicographic sums of chains over a finite poset $P_0$ of interest, and we obtain some general results on the linearization-count and sign-imbalance of these objects.