A113247 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have the same parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have the same parity where inv is the inversion number.

1, 1, 2, 4, 16, 64, 384, 2544, 20352, 181632, 1816320, 19960320, 239523840, 3113533440, 43589468160, 653837506560, 10461400104960, 177843719208960, 3201186945761280, 60822550297313280, 1216451005946265600

Offset

0,3

Comments

a(2n) and a(2n+1) are both divisible by 2^n n! a(2n) = 2n a(2n-1) The number of pi in S_n such that maj pi and maj pi^(-1) are both even is exactly half of a(n)

Links

Table of n, a(n) for n=0..20.

H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, preprint, 2005.

H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, Advances in Applied Mathematics, Volume 39, Issue 2, August 2007, Pages 269-281.

Formula

a(2n) = 2 n^2 a(2n-2) + 2 n (n-1) b(2n-2) and a(2n+1) = 2 n (n+1) a(2n-1) + 2 n^2 b(2n-1) where b(n) is sequence A113248

Example

a(3)=4 because the following 4 permutations in S_3 have the same parity for their major index and the major index of their inverse (and in fact are equal to their inverse): 123, 213, 321, 132.

Crossrefs

Cf. A113248.

Sequence in context: A155543 A151371 A001900 * A280132 A138870 A153992

Adjacent sequences: A113244 A113245 A113246 * A113248 A113249 A113250

Keyword

nonn

Author

Bruce E. Sagan, Oct 20 2005

Status

approved