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

%I A267324 #57 Aug 12 2020 11:42:14
%S A267324 0,0,0,0,0,32,288,2448,22080,216000,2298240,26530560,330946560,
%T A267324 4441651200,63866880000,980037273600,15990989414400,276529539686400,
%U A267324 5052853757952000,97290972979200000,1969085601939456000,41794695550992384000,928395406320205824000
%N A267324 Number of elements of S_n with strategic pile of size 4.
%C A267324 Strategic pile is defined in A267323.
%C A267324 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
%H A267324 K. L. M. Adamyk, E. Holmes, G. R. Mayfield, D. J. Moritz, M. Scheepers, B. E. Tenner, H. C. Wauck, <a href="http://arxiv.org/abs/1410.2353">Sorting Permutations: Games, Genomes, and Cycles</a>, arXiv:1410.2353 [math.CO], 2014-2017.
%H A267324 Marisa Gaetz, Bethany Flanagan, Marion Scheepers, Meghan Shanks, <a href="https://arxiv.org/abs/1811.11937">Quantifying CDS Sortability of Permutations by Strategic Pile Size</a>, arXiv:1811.11937 [math.CO], 2018.
%H A267324 M. Gaetz, B. Molokach, M. Scheepers, and M. Shanks, <a href="https://math.boisestate.edu/reu/publications/quantifying-cds-sortability.pdf">Quantifying CDS Sortability of Permutations Using Strategic Piles</a>
%F A267324 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
%e A267324 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}.
%t A267324 a[n_] := If[n<6, 0, 2(n-5)(n^2-5n+10) Pochhammer[3, n-6]];
%t A267324 Array[a, 23] (* _Jean-François Alcover_, Dec 12 2018 *)
%Y A267324 Cf. A267323 gives the corresponding sequence for strategic piles of size 3, A267391 for size 5, and A281259 for size 6.
%K A267324 nonn
%O A267324 1,6
%A A267324 _Marion Scheepers_, Jan 13 2016
%E A267324 Typo for a(8) corrected by _Marion Scheepers_, Jun 26 2016