A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions

Abstract

Let ${\mathcal{I}}_n^B$ be the involutions of the hyperoctahedral group ${\mathfrak{B}}_n$, and let $\des^B$ denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity \[ \sum_{n\ge0}\frac{\mathcal F_n(p,t)\,z^n}{(1-t)^{n+1}} =\sum_{m\ge0} \frac{(1+pz)^m\,t^m}{(1-pz)^{m+1}(1-z^2)^{m(m+1)}}, \quad \mathcal F_n(p,t)=\sum_{π\in\mathcal I_n^B}p^{\fixB(π)}t^{\des^B(π)}. \] Extracting the stratum with $j$ two-cycles and $f$ fixed positions yields a one-parameter deformation of the fixed-point-free Worpitzky series of Wan, Gao, Li and Yang. After the change of variables $x=t/(1+t{)}^2$, this deformation becomes a positive binomial transform. More precisely, if \[ P_j(x)=\sum_sD_{2j,s}x^s \] is the fixed-point-free $\gamma$-polynomial, then the transform coefficients $A_{j,r}(x)$ are determined by \[ \sum_{r\ge0}A_{j,r}(x)W^r =\sum_{s=0}^{j}D_{2j,s}x^s (1+4xW)^{2j-2s}(1+2W+4xW^2)^s, \] and the $\gamma$-polynomial of the $(j,f)$-stratum is \[ Φ_{j,f}(x)=\sum_{r=0}^{f}\binom fr A_{j,r}(x). \] This manifestly positive transform is the main structural result of the paper. As consequences, every fixed cycle-type stratum is $\gamma$-positive and ${\mathcal{F}}_n(p,t)$ is coefficientwise $\gamma$-positive in the fixed-point variable $p$. The cases $f=0$ and $p=1$ recover, respectively, the fixed-point-free theorem of Wan--Gao--Li--Yang and the all-involution theorem of Cao--Liu. We also give explicit formulas for the first binomial layers and for the strata with one and two two-cycles.

12 pages