A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type B Involutions
Abstract
Let ℐnB be the involutions of the hyperoctahedral group Bn, and let desB denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity ∑n≥0(ℱn(p,t) zn)/((1−t)n+1) =∑m≥0 ((1+pz)m tm)/((1−pz)m+1(1−z2)m(m+1)), ℱn(p,t)=∑π∈ℐnBpfixB(π)tdesB(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 Pj(x)=∑sD2j,sxs is the fixed-point-free γ-polynomial, then the transform coefficients Aj,r(x) are determined by ∑r≥0Aj,r(x)Wr =∑s=0jD2j,sxs (1+4xW)2j−2s(1+2W+4xW2)s, and the γ-polynomial of the (j,f)-stratum is Φj,f(x)=∑r=0fbinom fr Aj,r(x). This manifestly positive transform is the main structural result of the paper. As consequences, every fixed cycle-type stratum is γ-positive and ℱn(p,t) is coefficientwise γ-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.
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BibTeX
@misc{zeng2026fixedpoint,
title = {{A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions}},
author = {Jiang Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04922},
eprint = {2609.04922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04922},
}
RIS
TY - UNPB
TI - A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type B Involutions
AU - Jiang Zeng
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04922
UR - https://arxiv.org/abs/2609.04922
AB - Let ℐ_n^B be the involutions of the hyperoctahedral group 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 ∑_n≥0(ℱ_n(p,t) z^n)/((1−t)^n+1) =∑_m≥0 ((1+pz)^m t^m)/((1−pz)^m+1(1−z^2)^m(m+1)), ℱ_n(p,t)=∑_π∈ℐ_n^Bp^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)=∑_sD_2j,sx^s is the fixed-point-free γ-polynomial, then the transform coefficients A_j,r(x) are determined by ∑_r≥0A_j,r(x)W^r =∑_s=0^jD_2j,sx^s (1+4xW)^2j−2s(1+2W+4xW^2)^s, and the γ-polynomial of the (j,f)-stratum is Φ_j,f(x)=∑_r=0^fbinom 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 γ-positive and ℱ_n(p,t) is coefficientwise γ-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.
ER -