A302903 Number of permutations of [2n+1] having exactly n alternating descents.

1, 2, 36, 1196, 76840, 7570716, 1085246904, 211595659320, 53984412657360, 17440458896525180, 6960292943873805976, 3362366089440205308072, 1933633403768889597292336, 1305355624659052356741634136, 1022196734801743485304805455920, 919035074839303194470918726496240

Offset: 0

Author

Alois P. Heinz, Apr 15 2018

Keywords

• nonn

Comments

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(1) = 2: 123, 321.
a(2) = 36: 12345, 12543, 13542, 14325, 14532, 15324, 15423, 21354, 21453, 23541, 24315, 24531, 25314, 25413, 31254, 31452, 32145, 32451, 34215, 34521, 35214, 35412, 41253, 41352, 42135, 42351, 43125, 45213, 45312, 51243, 51342, 52134, 52341, 53124, 54123, 54321.
		

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 (odd 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+1, 0), x, n+1):
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[2n + 1, 0], x, n + 1];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) = A145876(2n+1,n+1) = A302905(2n+1).

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