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%I #25 Jan 23 2019 12:18:01
%S 0,0,0,3,12,66,432,3240,27360
%N The number of permutations in S_n with strategic pile of size 3.
%C The strategic pile of permutation P=[a_1,...,a_n] is obtained from the disjoint cycle decomposition of the composition of the cycles (a_n, ..., a_1,0)(0,1, 2, ..., n). If 0 and n are not in the same cycle, the strategic pile of P is empty. Else, the terms appearing from n to 0, not including n or 0, in the cycle (n, ..., 0, ...) is the strategic pile of P.
%C The strategic pile of P=[3,2,4,1] is {1, 2, 3} which has size 3 because: (1,4,2,3,0)(0,1,2,3,4) = ( 4, 1, 3, 2, 0).
%H 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 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.
%e a(4) = 3 because [3,2,4,1], [2,4,1,3] and [4,1,3,2] are the only elements of S_4 that each has a strategic pile of size 3.
%Y A267324 gives the corresponding sequence for strategic piles of size 4.
%Y A267391 gives the corresponding sequence for strategic piles of size 5.
%K nonn,more
%O 1,4
%A _Marion Scheepers_, Jan 13 2016
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