Factorizations of an n-cycle into two n-cycles

On the cycle structure of the product of random maximal cycles

“…For odd N , the result is equivalent to a well-known, but not at all obvious, fact that there are 2(N −1)! N +1 ways to factor a given maximal cycle into a product of two maximal cycles; see for instance [4] and the references therein. In general, the equations ( 10), ( 15) imply the following.…”

Quantifying CDS sortability of permutations by strategic pile size

“…Example 4.6. The permutation π = [5 4 6 3 2 1] has strategic pile SP(π) = {1, 3, 4, 5}, and SP * (π) = (1,3,5,4)…”

“…, a m ], with no parts of size one, with an even number of even parts, and with 1≤i≤m a i ≤ k. Figure 4. A τ -graph with cycle structure [3,3] Example 5.13. Consider the graph in Figure 4, which has cycle structure [3,3].…”

“…A τ -graph with cycle structure [3,3] Example 5.13. Consider the graph in Figure 4, which has cycle structure [3,3]. Notice that this cycle structure satisfies the conditions described in Remark 5.12, which are necessary conditions for a cycle structure to correspond to a τ -graph.…”

“…In section 2, we determine the number of elements in S n that have the maximum size strategic pile among all elements of S n . This counting problem reduces to a variation of the cycle factoring problem for S n , studied previously, and on the cycle factoring results of [2] and [3]. In section 3, we investigate how prevalent it is for permutations in S n to have strategic piles of cardinality k. As a result of this work we develop formulas in closed form that produce the terms of the integer sequences A267323, A267324 and A267391 in [10].…”