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

作者：Chen, Ricky X. F.^*

来源：JOURNAL OF ALGEBRAIC COMBINATORICS, 2023, 57(4): 1163-1171.

DOI：10.1007/s10801-023-01221-x

• Factorization of a permutation
• Permutation products
• Cycle
• Bijective function
• Top connection coefficient

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 Bona. 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 ? (depending on D and E). We prove that for arbitrary D and E with few exceptions, the number of cyclic permutations ? such that D ? ? has exactly ?- 1 E-label free cycles is at least 1/2 that of ? for D ? ? to have 0 E-label free cycles, where 1/2 is best possible. An even more general result is also conjectured.