A320982 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of eight.

0, 1, 9, 2061, 20736, 1507293, 17118482, 849597609, 10896848134, 459634150710, 6622740258098, 260269850813548, 4183025733598532, 160312786582715100, 2852747036627655519, 109305622312149855474, 2138487419350175956196, 83104784124134380097202

Offset: 8

Author

Alois P. Heinz, Oct 25 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..451

Crossrefs

Column k=8 of A262125.

Cf. A262132, A262133.

Programs

Maple

b:= proc(u, o, c) option remember; `if`(c<0 or c>8, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..8))(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-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 8):
seq(a(n), n=8..30);

Formula

a(n) = A262133(n) - A262132(n).