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arXiv · 2106.06021

Involutions under Bruhat order and labeled Motzkin Paths

Michael Coopman·Zachary Hamaker
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Abstract

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola M\'esz\'aros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner game.

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Michael Coopman, Zachary Hamaker. 2021-06-10. Involutions under Bruhat order and labeled Motzkin Paths. https://arxiv.org/abs/2106.06021

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