The Cycle Descent Statistic on Permutations

Jun Ma,Yeong-Nan Yeh,Shi-Mei Ma,Xu Zhu

doi:10.37236/5673

Abstract

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.

Highlights

Let Sn be the symmetric group of all permutations of [n], where [n] = {1, 2, . . . , n}
For any negative cycle descent permutation (π, φ) of [n], we have com (Γn(π, φ)) = cyc (π), ver (Γn(π, φ)) = fix (π), and neg (π, φ) if (1, 1) and its partner are in the same row, down (Γn(π, φ)) =
Let Γ|Dn denote the restriction of Γn on the set of negative cycle descent derangements of [n]

Summary

Introduction

The permutation π = 3142765 has the cycle decomposition (1342)(57)(6), so cyc (π) = 3, exc (π) = 3 and fix (π) = 1. Ksavrelof and Zeng [8] constructed bijective proofs of 1 and the following formula: xexc (π)(−1)cyc (π) = −x − x2 − · · · − xn−1 Their bijection leads to a refinement of the above identity: xexc (π)(−1)cyc (π) = −xn−i, π∈Dn,i where Dn,i is the set of derangements π of [n] such that π(n) = i. The above formulas can be proved by taking x = 1 in Theorem 1 of Section 2 Motivated by the these formulas, we shall study the cycle descent statistic of permutations. We present the main results of this paper and collect some notation and definitions that will be needed in the rest of the paper

Definitions and main results

Proof of the explicit formula 2 in Theorem 1

Proof of the explicit formula 3 in Theorem 1 n

Proof of the recurrence relation 4

Proof of the recurrence relation 5

Proof of Theorem 9