The $1/k$-Eulerian Polynomials of Type $B$

Shi-Mei Ma,Yeong-Nan Yeh,Jean Yeh,Jun Ma

doi:10.37236/9313

Abstract

In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.

Highlights

Throughout this paper, we always let k be a fixed positive number
It is well known that the statistics des (π), asc (π) and exc (π) are equidistributed over Sn, and their common enumerative polynomial is the Eulerian polynomial An(x), i.e., An(x) =
According to [5, Theorem 3.15], the statistics des B(w) and wexc (w) have the same distribution over Bn, and their common enumerative polynomial is the Eulerian polynomial of type B: Bn(x) =

Summary

Introduction

Throughout this paper, we always let k be a fixed positive number. Following Savage and Viswanathan [23], the 1/k-Eulerian polynomials A(nk)(x) are defined by. Asc (π), exc (π)) denote the number of descents It is well known that the statistics des (π), asc (π) and exc (π) are equidistributed over Sn, and their common enumerative polynomial is the Eulerian polynomial An(x), i.e., An(x) =. Exc (w), fix (w), single (π)) denote the number of weak excedances According to [5, Theorem 3.15], the statistics des B(w) and wexc (w) have the same distribution over Bn, and their common enumerative polynomial is the Eulerian polynomial of type B: Bn(x) =. According to [11, Theorem 3.2], the generating function of dBn (x) is given as follows: dBn (1 − x)ez. The type B 1/k-Eulerian polynomials Bn(k)(x) and the type B 1/k-derangement polynomials dBn (x; k) are defined by using the following generating functions: Bn(k).

Main results

Proof of Theorem 4

Proof of Theorem 9