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

1,6

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

Links

Table of n, a(n) for n=1..23.

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

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

Example

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}.

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 *)

Crossrefs

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

Sequence in context: A316881 A317609 A251782 * A197523 A337418 A283546

Adjacent sequences: A267321 A267322 A267323 * A267325 A267326 A267327

Keyword

nonn

Author

Marion Scheepers, Jan 13 2016

Extensions

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

Status

approved