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arXiv · 2512.12275

Counting permutations by alternating runs via Hetyei-Reiner trees

Qiongqiong Pan·Yunze Wang·Jiang Zeng
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Abstract

The generating polynomial of permutations of size $n$, counted by the number of alternating runs, has a root at $-1$ of multiplicity $\lfloor (n-2)/2 \rfloor$ for all $n \ge 2$. This result can be derived by combining the David--Barton formula for Eulerian polynomials with the Foata--Sch\"utzenberger $\gamma$--decomposition. More recently, B\'ona 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 B\'ona's quotient polynomial. Moreover, we extend our analysis to analogous results for permutations of types~$B$ and~$D$. As a by--product of our bijective framework, we also obtain combinatorial proofs of David--Barton--type identities for permutations of types~$A$ and~$B$.

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Qiongqiong Pan, Yunze Wang, Jiang Zeng. 2025-12-13. Counting permutations by alternating runs via Hetyei-Reiner trees. https://arxiv.org/abs/2512.12275

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