A267324 Number of elements of S_n with strategic pile of size 4.

0, 0, 0, 0, 0, 32, 288, 2448, 22080, 216000, 2298240, 26530560, 330946560, 4441651200, 63866880000, 980037273600, 15990989414400, 276529539686400, 5052853757952000, 97290972979200000, 1969085601939456000, 41794695550992384000, 928395406320205824000

Offset: 1

Author

Marion Scheepers, Jan 13 2016

Keywords

• nonn

Comments

Strategic pile is defined in A267323.

The formula given below is a specific instance of the formula that will appear in "Quantifying CDS Sortability of Permutations Using Strategic Piles", see link. - Marisa Gaetz, Jan 18 2017

Examples

			P = [6,4,2,5,3,1] has strategic pile of size 4: The composition of cycles (0,1,3,5,2,4,6)(0,1,2,3,4,5,6) is (0,3,6,1,4,2,5) = (6,1,4,2,5,0,3) and thus the strategic pile of P is {1,2,4,5}.
		

Links

• K. L. M. Adamyk, E. Holmes, G. R. Mayfield, D. J. Moritz, M. Scheepers, B. E. Tenner, H. C. Wauck, Sorting Permutations: Games, Genomes, and Cycles, arXiv:1410.2353 [math.CO], 2014-2017.
• Marisa Gaetz, Bethany Flanagan, Marion Scheepers, Meghan Shanks, Quantifying CDS Sortability of Permutations by Strategic Pile Size, arXiv:1811.11937 [math.CO], 2018.
• M. Gaetz, B. Molokach, M. Scheepers, and M. Shanks, Quantifying CDS Sortability of Permutations Using Strategic Piles

Crossrefs

Cf. A267323 gives the corresponding sequence for strategic piles of size 3, A267391 for size 5, and A281259 for size 6.

Programs

Mathematica

a[n_] := If[n<6, 0, 2(n-5)(n^2-5n+10) Pochhammer[3, n-6]];
Array[a, 23] (* Jean-François Alcover, Dec 12 2018 *)

Formula

a(n) = (n-4)!*(6*binomial(n-5,3) + 16*binomial(n-5,2) + 16*binomial(n-5,1)) for n>5. - Marisa Gaetz, Jan 18 2017

Extensions

Typo for a(8) corrected by Marion Scheepers, Jun 26 2016