Counting permutations by alternating runs via Hetyei-Reiner trees

Qiongqiong Pan; Yunze Wang; Jiang Zeng

Paper

Counting permutations by alternating runs via Hetyei-Reiner trees

Abstract

The generating polynomial of permutations of size

, counted by the number of alternating runs, has a root at

of multiplicity

for all

. This result can be derived by combining the David--Barton formula for Eulerian polynomials with the Foata--Schützenberger

--decomposition. More recently, Bóna gave a group--action proof of this phenomenon. In this paper, we present an alternative approach based on the Hetyei--Reiner action on binary trees, which leads to a new combinatorial interpretation of Bóna's quotient polynomial. Moreover, we extend our analysis to analogous results for permutations of types~

and~

. As a by--product of our bijective framework, we also obtain combinatorial proofs of David--Barton--type identities for permutations of types~

and~