A135410 Number of permutations of {1,2,3,...,n} each with the same up-down signature as its inverse permutation.

1, 1, 2, 4, 10, 26, 80, 272, 1076, 4848, 24832, 142340, 902440, 6257264, 47042924, 380800060, 3300070116, 30472353884, 298595442140

Offset

0,3

Comments

An "up-down signature" of the permutation (p(1),p(2),p(3),...,p(n)) is the sequence of (n-1) signs (+ or -) where the k-th sign in the sequence is the sign of (p(k+1)-p(k)). All permutations of {1,2,3,...,n} which are their own inverses obviously have the same up-down signature as their inverse permutations.

The terms were calculated by R. J. Mathar (see link).

Links

Table of n, a(n) for n=0..18.

R. J. Mathar, C++ program

Example

The up-down signature of the permutation (4,2,6,3,5,1) is -+-+-. This permutation's inverse permutation is (6,2,4,1,5,3). The inverse permutation also has the up-down signature -+-+-. So these two permutations, among others, are included in the count for n = 6.

Crossrefs

Cf. A000085.

Sequence in context: A003239 A195924 A116673 * A148103 A148104 A179981

Adjacent sequences: A135407 A135408 A135409 * A135411 A135412 A135413

Keyword

nonn,more

Author

Leroy Quet, Dec 10 2007

Extensions

a(13)-a(14) from Alois P. Heinz, Oct 18 2013

a(15)-a(18) from Martin Ehrenstein, Jul 30 2023

Status

approved