2021/06/10 by Coopman, Michael, Hamaker, Zachary
#05A19 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.06021
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 Mészáros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner game.