TY - JOUR

T1 - Signed Hultman numbers and signed generalized commuting probability in finite groups

AU - Shwartz, Robert

AU - Levit, Vadim E.

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/6

Y1 - 2022/6

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

UR - http://www.scopus.com/inward/record.url?scp=85122157426&partnerID=8YFLogxK

U2 - 10.1007/s10801-021-01089-9

DO - 10.1007/s10801-021-01089-9

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AN - SCOPUS:85122157426

SN - 0925-9899

VL - 55

SP - 1171

EP - 1197

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -