On products of permutations with the most uncontaminated cycles by designated labels

• Published in: Journal of algebraic combinatorics Vol. 57; no. 4; pp. 1163 - 1171
• Main Author: Chen, Ricky X. F.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2023; Springer Nature B.V
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Labels; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Permutations; Bijective function; Top connection coefficient; Factorization of a permutation; 05E16; 05A05; Permutation products; Cycle
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-023-01221-x

There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of Bóna. This paper is concerned with a problem in that direction. Let D be a permutation on the set [ n ] = { 1 , 2 , … , n } and E ⊂ [ n ] . Suppose the maximum possible number of cycles uncontaminated by the E -labels in the product of D and a cyclic permutation on [ n ] is θ (depending on D and E ). We prove that for arbitrary D and E with few exceptions, the number of cyclic permutations γ such that D ∘ γ has exactly θ - 1 E -label free cycles is at least 1/2 that of γ for D ∘ γ to have θ E -label free cycles, where 1/2 is best possible. An even more general result is also conjectured.