A320977 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of three.

0, 1, 4, 57, 311, 3857, 27589, 355751, 3185258, 44435241, 479927081, 7311147055, 92603527295, 1540826943617, 22398016781688, 405865410503155, 6658173453910401, 130871330990044897, 2390845313044048301, 50759828657781860167, 1021623022236754343662

Offset: 3

Author

Alois P. Heinz, Oct 25 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..458

Crossrefs

Column k=3 of A262125.

Cf. A262126, A262128.

Programs

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-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 3):
seq(a(n), n=3..30);

Formula

a(n) = A262128(n) - A262126(n).