A320978 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of four.

0, 1, 5, 123, 778, 14126, 111811, 1957924, 18846572, 342736488, 3904186779, 75626480215, 996651530998, 20739117286767, 310506509296055, 6952893022274589, 116561453850733664, 2805243168612301480, 52046467944203248643, 1342921799608903158968

Offset: 4

Author

Alois P. Heinz, Oct 25 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..455

Crossrefs

Column k=4 of A262125.

Cf. A262128, A262129.

Programs

Maple

b:= proc(u, o, c) option remember; `if`(c<0 or c>4, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..4))(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, 4):
seq(a(n), n=4..30);

Formula

a(n) = A262129(n) - A262128(n).