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Poset edge densities, nearly reduced words, and barely set-valued tableaux. (English) Zbl 1391.05269
Summary: In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young’s lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.

05E10 Combinatorial aspects of representation theory
06A07 Combinatorics of partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C08 Hecke algebras and their representations
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