A250259 -id:A250259 - cp's OEIS frontend

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

Alois P. Heinz, Nov 15 2014

A(n,0) = A(n,k) for k>=n-1 and n>0.

			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, ...

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

Columns k=1-10 give: A000012, A000111, A249402, A250259, A250260, A250262, A250263, A250264, A250265, A250266.

A(n+3,n+1) = A028387(n).

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);

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 *)

A250260 The number of 5-alternating permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 29, 133, 412, 1041, 2300, 22991, 170832, 822198, 3114489, 10006375, 141705439, 1457872978, 9522474417, 48094772656, 202808749375, 3716808948931, 48860589990687, 403131250565618, 2545098156762649, 13287626090593750

Offset: 0

R. J. Mathar, Nov 15 2014

nonn

A sequence a(1), a(2),... is called k-alternating if a(i) > a(i+1) iff i=1 (mod k).

Alois P. Heinz, Table of n, a(n) for n = 0..500
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, page 17.

Cf. A065619 (2-alternating), A249402 (3-alternating), A250259 (4-alternating).

Column k=5 of A250261.

# dowupP defined in A250259.
A250260 :=proc(n)
    downupP(n,4) ;
end proc:
seq(A250260(n),n=0..20) ;
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, 5)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 5)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 15 2014

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, If[t == 1, Sum[b[u-j, o+j-1, Mod[t+1, 5]], {j, 1, u}], Sum[b[u+j-1, o-j, Mod[t+1, 5]], {j, 1, o}]]]; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)

cp's OEIS Frontend

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.

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