A302904 Number of permutations of [2n] having exactly n-1 alternating descents.

1, 1, 7, 182, 8699, 704834, 84889638, 14322115212, 3216832016019, 928559550102410, 334876925319944690, 147563833511292247796, 78009671642511668089822, 48728981875112003682759892, 35506576774281843111748649644, 29848802048200930275501944893080

Offset: 0

Author

Alois P. Heinz, Apr 15 2018

Keywords

• nonn

Comments

a(0) = 1 by convention.

Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

Examples

			a(2) = 7: 1234, 1432, 2431, 3214, 3421, 4213, 4312.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.

Crossrefs

Bisection (even part) of A302905.

Cf. A145876.

Programs

Maple

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)))
    end:
a:= n-> coeff(b(2*n, 0), x, n):
seq(a(n), n=0..20);

Mathematica

b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
    Sum[b[o + j - 1, u - j] x, {j, 1, u}] +
    Sum[b[o - j, u - 1 + j],   {j, 1, o}]]];
a[n_] := Coefficient[b[2 n, 0], x, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) = A145876(2n,n) = A302905(2n) for n > 0.

a(n) ~ sqrt(3) * 2^(2*n + 1) * n^(2*n) / (sqrt(5) * exp(2*n)). - Vaclav Kotesovec, Apr 29 2018