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

Shi-Mei Ma, Jun Ma, Jean Yeh, Yeong-Nan Yeh
2020 Electronic Journal of Combinatorics  
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
more » ... In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
doi:10.37236/9313 fatcat:foscbtrmivgm5pnc3gs7hkvdza