Odd permutations are nicer than even ones

International audienceWe give simple combinatorial proofs of some formulas for the number of factorizations of permutations in S n as a product of two n-cycles, or of an n-cycle and an (n−1)-cycle. ... The parameter number of cycles plays a central role in the algebraic theory of the symmetric group, however there are very few results giving a relationship between the number of cycles of two permutations and that of their product. ... The first results on the subject go back to Ore, Bertram, Stanley (see [13], [1] and [15]), who proved some existence theorems. These results allowed to obtain ..

Explicit formulas for hypermaps and maps

The study of hypermaps and maps is ubiquitous, as they are closely connected with geometry, mathematical physics, free probability and algebra. A novel recursion generalizing a fundamental identity of Frobenius which enumerates factorizations of a permutation in group algebra theory has been discovered by the author recently. Here we apply the recursion to study hypermaps as well as maps and obtain a plethora of results in a unified way. For instance, we succinctly provide a short proof of the celebrated Harer-Zagier formula and present a general explicit formula for one-face hypermaps. As special cases of the latter, we obtain simple explicit formulas for the numbers of ways of expressing a long cycle as a product of a permutation of cycle-type $[1^p, n-p]$ and, respectively, $[p, n-p]$ and a permutation with $m$ cycles for any $p$ and $m$. To the best of our knowledge, only the cases for $p=0,\, 1$ are known before.Comment: corrected the formula in Theorem 3.

A versatile combinatorial approach of studying products of long cycles in symmetric groups

In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial proofs, including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new results as well. In particular, for the first time, given a partition of a set, we obtain an explicit formula for the number of pairs of long cycles on the set such that the product of the long cycles does not mix the elements from distinct blocks of the partition and has an independently prescribed number of cycles for each block of elements. As applications, we obtain new explicit formulas concerning factorizations of any even permutation into long cycles and the first nontrivial explicit formula for computing strong separation probabilities solving an open problem of Stanley (2010).Comment: 12 pages, a draft extended abstract, comments are welcome. arXiv admin note: substantial text overlap with arXiv:1909.13388; text overlap with arXiv:1910.0102

On products of long cycles: short cycle dependence and separation probabilities

We present various results on multiplying cycles in the symmetric group. One result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on n symbols as a product of an n-cycle and an (n - 1)-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths. Another application is related to the recent notion of separation probabilities for permutations introduced by Bernardi, Du, Morales and Stanley (2014)