Signed Hultman numbers and signed generalized commuting probability in finite groups

Robert Shwartz; Vadim E. Levit

doi:10.1007/s10801-021-01089-9

Signed Hultman numbers and signed generalized commuting probability in finite groups

Robert Shwartz, Vadim E. Levit

Department of Mathematics

Ariel University

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a finite group. Let π be a permutation from S_n. We study the distribution of probabilities of equalitya1a2⋯an-1an=aπ1ϵ1aπ2ϵ2⋯aπn-1ϵn-1aπnϵn, when π varies over all the permutations in S_n, and ϵ_i varies over the set { + 1 , - 1 }. By [7], the case where all ϵ_i are + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting ϵ_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.

Original language	American English
Pages (from-to)	1171-1197
Number of pages	27
Journal	Journal of Algebraic Combinatorics
Volume	55
Issue number	4
DOIs	https://doi.org/10.1007/s10801-021-01089-9
State	Published - 1 Jun 2022

Keywords

Breakpoint graph
Commuting probability
Finite group
Signed Hultman number
Signed permutation

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-021-01089-9

Cite this

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title = "Signed Hultman numbers and signed generalized commuting probability in finite groups",

abstract = "Let G be a finite group. Let π be a permutation from Sn. We study the distribution of probabilities of equalitya1a2⋯an-1an=aπ1ϵ1aπ2ϵ2⋯aπn-1ϵn-1aπnϵn, when π varies over all the permutations in Sn, and ϵi varies over the set \{ + 1 , - 1 \}. By [7], the case where all ϵi are + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting ϵi 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.",

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doi = "10.1007/s10801-021-01089-9",

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AU - Levit, Vadim E.

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N2 - Let G be a finite group. Let π be a permutation from Sn. We study the distribution of probabilities of equalitya1a2⋯an-1an=aπ1ϵ1aπ2ϵ2⋯aπn-1ϵn-1aπnϵn, when π varies over all the permutations in Sn, and ϵi varies over the set { + 1 , - 1 }. By [7], the case where all ϵi are + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting ϵi 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.

AB - Let G be a finite group. Let π be a permutation from Sn. We study the distribution of probabilities of equalitya1a2⋯an-1an=aπ1ϵ1aπ2ϵ2⋯aπn-1ϵn-1aπnϵn, when π varies over all the permutations in Sn, and ϵi varies over the set { + 1 , - 1 }. By [7], the case where all ϵi are + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting ϵi 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.

KW - Breakpoint graph

KW - Commuting probability

KW - Finite group

KW - Signed Hultman number

KW - Signed permutation

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DO - 10.1007/s10801-021-01089-9

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JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -

Signed Hultman numbers and signed generalized commuting probability in finite groups

Abstract

Keywords

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