Permutation statistics on involutions

2004/12/11 by Mark Dukes, W. M. B. Dukes, Dukes, W. M. B.
Agricultural and Biological Sciences · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.math/0412222

arxiv created 2004/12/11 · openalex publication_date 2004/12/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we look at polynomials arising from statistics on the classes of involutions, I_n, and involutions with no fixed points, J_n, in the symmetric group. Our results are motivated by F. Brenti's conjecture which states that the Eulerian distribution of I_n is log-concave. Symmetry of the generating functions is shown for the statistics des,maj and the joint distribution (des,maj). We show that exc is log-concave on I_n, inv is log-concave on J_n and des is partially unimodal on both I_n and J_n. We also give recurrences and explicit forms for the generating functions of the inversions statistic on involutions in Coxeter groups of types B_n and D_n. Symmetry and unimodality of inv is shown on the subclass of signed permutations in D_n with no fixed points. In light of these new results, we present further conjectures at the end of the paper.

Related

• Involution Statistics in Finite Coxeter Groups
• Permutation Statistics in Conjugacy Classes of the Symmetric Group
• Joint extremes of inversions and descents of random permutations
• Involution factorizations of Ewens random permutations
• The Eulerian distribution on involutions is indeed γ-positive