# Signed Hultman Numbers and Signed Generalized Commuting Probability in Finite Groups

## TL;DR

This paper extends previous work on permutation equalities in finite groups by including sign variations, analyzing the distribution of probabilities related to signed permutations and their connection to signed Hultman numbers.

## Contribution

It generalizes prior results by allowing epsilon{i} to be -1, describing the spectrum of signed permutation probabilities in finite groups, and linking it to signed Hultman numbers.

## Key findings

- Spectrum of signed permutation probabilities characterized
- Connection established between probabilities and signed Hultman numbers
- Generalization of previous results to include negative signs

## Abstract

Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led to a close connection to Hultman numbers. In this paper we generalize the results, permitting epsilon{i} to be -1. We describe the spectrum of the probabilities of signed permutation equalities in a finite group G. This spectrum turns out to be closely related to the partition of 2^{n}*n! into a sum of the corresponding signed Hultman numbers.