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A250261 Number A(n,k) of permutations p of [n] such that p(i) > p(i+1) iff i = 1 + k*m for some m >= 0; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A250261 #24 Jan 13 2019 18:52:27
%S A250261 1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,2,1,4,1,1,1,2,5,1,5,1,1,1,2,3,16,
%T A250261 1,6,1,1,1,2,3,11,61,1,7,1,1,1,2,3,4,40,272,1,8,1,1,1,2,3,4,19,99,
%U A250261 1385,1,9,1,1,1,2,3,4,5,78,589,7936,1,10,1,1,1,2,3,4,5,29,217,3194,50521,1,11
%N A250261 Number A(n,k) of permutations p of [n] such that p(i) > p(i+1) iff i = 1 + k*m for some m >= 0; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A250261 A(n,0) = A(n,k) for k>=n-1 and n>0.
%H A250261 Alois P. Heinz, <a href="/A250261/b250261.txt">Antidiagonals n = 0..140, flattened</a>
%H A250261 J. M. Luck, <a href="https://arxiv.org/abs/1309.7764">On the frequencies of patterns of rises and falls</a>, arXiv:1309.7764, 2013
%H A250261 A. Mendes and J. Remmel, Generating functions from symmetric functions, Preliminary version of book, available from <a href="http://math.ucsd.edu/~remmel/">Jeffrey Remmel's home page</a>
%H A250261 R. P. Stanley, <a href="https://arxiv.org/abs/0912.4240">A survey of alternating permutations</a>, arXiv:0912.4240, 2009
%e A250261 Square array A(n,k) begins:
%e A250261   1, 1,    1,   1,   1,   1,  1, 1, 1, ...
%e A250261   1, 1,    1,   1,   1,   1,  1, 1, 1, ...
%e A250261   1, 1,    1,   1,   1,   1,  1, 1, 1, ...
%e A250261   2, 1,    2,   2,   2,   2,  2, 2, 2, ...
%e A250261   3, 1,    5,   3,   3,   3,  3, 3, 3, ...
%e A250261   4, 1,   16,  11,   4,   4,  4, 4, 4, ...
%e A250261   5, 1,   61,  40,  19,   5,  5, 5, 5, ...
%e A250261   6, 1,  272,  99,  78,  29,  6, 6, 6, ...
%e A250261   7, 1, 1385, 589, 217, 133, 41, 7, 7, ...
%p A250261 b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
%p A250261      `if`(t=1, add(b(u-j, o+j-1, irem(t+1, k), k), j=1..u),
%p A250261                add(b(u+j-1, o-j, irem(t+1, k), k), j=1..o)))
%p A250261     end:
%p A250261 A:= (n, k)-> b(0, n, 0, `if`(k=0, n, k)):
%p A250261 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A250261 b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, If[t == 1, Sum[ b[u-j, o+j-1, Mod[t+1, k], k], {j, 1, u}], Sum[ b[u+j-1, o-j, Mod[t+1, k], k], {j, 1, o}] ] ] ; A[n_, k_] := b[0, n, 0, If[k == 0, n, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 03 2015, after _Alois P. Heinz_ *)
%Y A250261 Columns k=1-10 give: A000012, A000111, A249402, A250259, A250260, A250262, A250263, A250264, A250265, A250266.
%Y A250261 A(n+3,n+1) = A028387(n).
%K A250261 nonn,tabl
%O A250261 0,10
%A A250261 _Alois P. Heinz_, Nov 15 2014