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Some results on counting linearizations of posets
Abstract
In section 1 we consider a 3tuple $S=(S,\preccurlyeq,E)$ 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 numbertheoretic result about this function, and use it in section 7 to recover a ringtheoretic identity of G. P. Hochschild. In section 2 we generalize a result of R. Stanley on the signimbalance of posets in which the lengths of all maximal chains have the same parity. In sections 36 we study the linearizationcount and signimbalance 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 linearizationcount and signimbalance of these objects.
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