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.
@article{10_37236_9313,
author = {Shi-Mei Ma and Jun Ma and Jean Yeh and Yeong-Nan Yeh},
title = {The {\(1/k\)-Eulerian} polynomials of type {\(B\)}},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {3},
doi = {10.37236/9313},
zbl = {1445.05005},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9313/}
}
TY - JOUR
AU - Shi-Mei Ma
AU - Jun Ma
AU - Jean Yeh
AU - Yeong-Nan Yeh
TI - The \(1/k\)-Eulerian polynomials of type \(B\)
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/9313/
DO - 10.37236/9313
ID - 10_37236_9313
ER -
%0 Journal Article
%A Shi-Mei Ma
%A Jun Ma
%A Jean Yeh
%A Yeong-Nan Yeh
%T The \(1/k\)-Eulerian polynomials of type \(B\)
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/9313/
%R 10.37236/9313
%F 10_37236_9313
Shi-Mei Ma; Jun Ma; Jean Yeh; Yeong-Nan Yeh. The \(1/k\)-Eulerian polynomials of type \(B\). The electronic journal of combinatorics, Tome 27 (2020) no. 3. doi: 10.37236/9313
