Cyclotomic factorization and fixed-point insertion in colored permutation groups
Abstract
We extend cyclotomic Euler--Mahonian factorization to colored permutation groups for two inversion statistics: Biagioli--Zeng length and the inversion number of the underlying unsigned permutation. At a primitive mth root and rank n=mk+i, each normalized enumerator is a Hadamard product of a rank-i colored enumerator and a weighted rank-k type-B enumerator. Separate color variables are retained. The type-B factors agree for the two statistics, yielding equidistribution modulo the rank jointly with descent number, major index and negative-entry number. We also determine their exact changes under positive fixed-point insertion and obtain rational kernels retaining descent and major index. These kernels mark arbitrary fixed-point numbers and give derangement expansions at every inversion weight. Signed specializations recover type-A and type-B identities and give colored flag-major refinements. Inverse statistics and color-sum subgroups admit related insertion formulas.
Related
- Permutation automorphism groups of cyclic codes I: cyclotomic association schemes
- Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
- Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- Intersection Orbital Graphs of Permutation Groups and Reconstruction of Simple Groups
BibTeX
@misc{zeng2026cyclotomic,
title = {{Cyclotomic factorization and fixed-point insertion in colored permutation groups}},
author = {Jiang Zeng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05848},
eprint = {2611.05848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05848},
}
RIS
TY - UNPB
TI - Cyclotomic factorization and fixed-point insertion in colored permutation groups
AU - Jiang Zeng
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05848
UR - https://arxiv.org/abs/2611.05848
AB - We extend cyclotomic Euler--Mahonian factorization to colored permutation groups for two inversion statistics: Biagioli--Zeng length and the inversion number of the underlying unsigned permutation. At a primitive mth root and rank n=mk+i, each normalized enumerator is a Hadamard product of a rank-i colored enumerator and a weighted rank-k type-B enumerator. Separate color variables are retained. The type-B factors agree for the two statistics, yielding equidistribution modulo the rank jointly with descent number, major index and negative-entry number. We also determine their exact changes under positive fixed-point insertion and obtain rational kernels retaining descent and major index. These kernels mark arbitrary fixed-point numbers and give derangement expansions at every inversion weight. Signed specializations recover type-A and type-B identities and give colored flag-major refinements. Inverse statistics and color-sum subgroups admit related insertion formulas.
ER -