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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. 10
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS A(n,0) = A(n,k) for k>=n-1 and n>0. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764, 2013 A. Mendes and J. Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, 2009 EXAMPLE Square array A(n,k) begins: 1, 1,    1,   1,   1,   1,  1, 1, 1, ... 1, 1,    1,   1,   1,   1,  1, 1, 1, ... 1, 1,    1,   1,   1,   1,  1, 1, 1, ... 2, 1,    2,   2,   2,   2,  2, 2, 2, ... 3, 1,    5,   3,   3,   3,  3, 3, 3, ... 4, 1,   16,  11,   4,   4,  4, 4, 4, ... 5, 1,   61,  40,  19,   5,  5, 5, 5, ... 6, 1,  272,  99,  78,  29,  6, 6, 6, ... 7, 1, 1385, 589, 217, 133, 41, 7, 7, ... MAPLE b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,      `if`(t=1, add(b(u-j, o+j-1, irem(t+1, k), k), j=1..u),                add(b(u+j-1, o-j, irem(t+1, k), k), j=1..o)))     end: A:= (n, k)-> b(0, n, 0, `if`(k=0, n, k)): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA 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 *) CROSSREFS Columns k=1-10 give: A000012, A000111, A249402, A250259, A250260, A250262, A250263, A250264, A250265, A250266. A(n+3,n+1) = A028387(n). Sequence in context: A137773 A175010 A107454 * A063669 A211005 A162154 Adjacent sequences:  A250258 A250259 A250260 * A250262 A250263 A250264 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Nov 15 2014 STATUS approved

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Last modified October 22 14:36 EDT 2018. Contains 316486 sequences. (Running on oeis4.)