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%I #24 Jan 13 2019 18:52:27
%S 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 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 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 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 A(n,0) = A(n,k) for k>=n-1 and n>0.
%H Alois P. Heinz, <a href="/A250261/b250261.txt">Antidiagonals n = 0..140, flattened</a>
%H 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 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 R. P. Stanley, <a href="https://arxiv.org/abs/0912.4240">A survey of alternating permutations</a>, arXiv:0912.4240, 2009
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 3, 1, 5, 3, 3, 3, 3, 3, 3, ...
%e 4, 1, 16, 11, 4, 4, 4, 4, 4, ...
%e 5, 1, 61, 40, 19, 5, 5, 5, 5, ...
%e 6, 1, 272, 99, 78, 29, 6, 6, 6, ...
%e 7, 1, 1385, 589, 217, 133, 41, 7, 7, ...
%p b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
%p `if`(t=1, add(b(u-j, o+j-1, irem(t+1, k), k), j=1..u),
%p add(b(u+j-1, o-j, irem(t+1, k), k), j=1..o)))
%p end:
%p A:= (n, k)-> b(0, n, 0, `if`(k=0, n, k)):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t 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 Columns k=1-10 give: A000012, A000111, A249402, A250259, A250260, A250262, A250263, A250264, A250265, A250266.
%Y A(n+3,n+1) = A028387(n).
%K nonn,tabl
%O 0,10
%A _Alois P. Heinz_, Nov 15 2014
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