Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.7 |

Department of Mathematics

Ariel University

Israel

Vadim E. Levit

Department of Computer Sciences

Ariel University

Israel

Avraham Goldstein

Borough of Manhattan Community College

The City University of New York

USA

Robert Shwartz

Department of Mathematics

Ariel University

Israel

**Abstract:**

Let *G* be a finite group. We investigate the distribution of the
probabilities of the permutation equality

*a*_{1}*a*_{2} ... *a*_{n-1}
*a*_{n} = *a*_{π1}
*a*_{π2} ... *a*_{πn-1} *a*_{πn}

as π varies over all the permutations in *S*_{n}.
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_{1}^{ω}(*G*), with

ω =
*a*_{1}*a*_{2} ... *a*_{n-1}
*a*_{n} *a*_{π1}^{-1}
*a*_{π2}^{-1} ...
*a*_{πn-1}^{-1}
*a*_{πn}^{-1}

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 Erdős and Turan in 1968 and by Gustafson in 1973.
Gustafson established a relation between the probability of
*a*_{1}, *a*_{2} ∈ *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.

(Concerned with sequence A164652.)

Received December 17 2015; revised versions received March 29 2017; October 31 2017.
Published in *Journal of Integer Sequences*, November 22 2017.

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