## Abstract

Let G be a finite group and π be a permutation from Sn. We investigate the distribution of the probabilities of the equality a_{1}a_{2}⋯a_{n}−1a_{n}=a_{π1}a_{π2}⋯a_{πn−1}a_{πn} when π varies over all the permutations in Sn. The probability Pr_{π}(G)=Pr(a_{1}a_{2}⋯a_{n−1}a_{n}=a_{π1}a_{π2}⋯a_{πn−1}a_{πn}) is identical to Pr^{ω} _{1}(G), with ω=a_{1}a_{2}…a_{n−1}a_{n-1}a_{n}a^{−1}_{πn}a^{−}_{πn-1}… a^{−}_{π2}a^{−}_{π1}, which was defined and studied by Das and Nath. The notion of commutativity degree, or the probability of a permutation equality a_{1}a_{2} = a_{2}a_{1}, for which n = 2 and π = ⟨2 1⟩ was introduced and assessed by Erdös and Turan in 1968 and by Gustafson in 1973. Gustafson established a relation between the probability of a1, a2 ∈ G commuting and the number of conjugacy classes in G. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of G, and compute probabilities of permutation equalities in terms of these parameters. It turns out that for a permutation π, the probability of its permutation equality depends only on the number c(Gr(π)) of alternating cycles in the cycle graph Gr(π) of π. The cycle graph of a permutation was introduced by Bafna and Pevzner, and the number of alternating cycles in it was introduced by Hultman. Hultman numbers are the numbers of different permutations with the same number of alternating cycles in their cycle graphs. We show that the spectrum of probabilities of permutation equalities in a generic finite group, as π varies over all the permutations in S_{n}, corresponds to partitioning n! as the sum of the corresponding Hultman numbers.

Original language | English |
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Article number | 17.10.7 |

Journal | Journal of Integer Sequences |

Volume | 20 |

Issue number | 10 |

State | Published - 22 Nov 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017, University of Waterloo. All rights reserved.

## Keywords

- Commuting probability
- Cycle graph of a permutation
- Finite group
- Hultman number
- Isoclinism