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 P. 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 L. Billera, L. Levine and K. Mészáros with a previous bijective proof by M. Watson. Our work sheds new light on the Ethiopian dinner game.
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