A262165 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 3.

1, 1, 2, 5, 19, 82, 454, 2795, 20346, 162613, 1469309, 14424200, 155842828, 1812646171, 22807141756, 306480808871, 4403059520043, 67100946088054, 1084001371054298, 18469410744415367, 331442882307143590, 6242679740272435021, 123215973021475320637

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..458

Formula

a(n) = A262163(n,3).

Maple

b:= proc(u, o, c) option remember; `if`(c<0 or c>3, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..3))(add(
       b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 3)))(b(0, n, 0)):
seq(a(n), n=0..25);

Mathematica

b[u_, o_, c_] := b[u, o, c] = If[c < 0 || c > 3, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, 3}]][Sum[b[u - j, o - 1 + j, c - 1], {j, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, o}]]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, 3]}]][b[0, n, 0]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Crossrefs

Column k=3 of A262163.

Sequence in context: A288911 A138911 A181513 * A219661 A179566 A107377

Adjacent sequences: A262162 A262163 A262164 * A262166 A262167 A262168

Keyword

nonn

Author

Alois P. Heinz, Sep 13 2015

Status

approved