Pattern expansions of permutation statistics

Ian Cavey; Hugh Dennin; Bridget Eileen Tenner

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arXiv · 2601.03918

Pattern expansions of permutation statistics

Ian Cavey·Hugh Dennin·Bridget Eileen Tenner

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Abstract

We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the higher moment statistics, generalizing a result of Berman and Tenner. We also give a combinatorial criterion for the positivity of pattern expansions. Using this criterion, we show that the pattern expansion of the number of reduced words of a permutation is positive and give an enumerative interpretation for the coefficients.

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Ian Cavey, Hugh Dennin, Bridget Eileen Tenner. 2026-01-07. Pattern expansions of permutation statistics. https://arxiv.org/abs/2601.03918

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