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
Examples
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, ...
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
Crossrefs
Programs
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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 *)
Comments