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.

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

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: A175010 A107454 A371212 * A063669 A306489 A319734

Adjacent sequences: A250258 A250259 A250260 * A250262 A250263 A250264

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 15 2014

Status

approved