A113248 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have opposite parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have opposite parity where inv is the inversion number.

0, 0, 2, 8, 56, 336, 2496, 19968, 181248, 1812480, 19956480, 239477760, 3113487360, 43588823040, 653836861440, 10461389783040, 177843708887040, 3201186759966720, 60822550111518720, 1216451002230374400

Offset: 0

Author

Bruce E. Sagan, Oct 20 2005

Keywords

• nonn

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 is even and maj pi^(-1) is odd is exactly half of a(n)

Examples

			a(3)=2 because the following 2 permutations in S_3 have opposite parity for their major index and the major index of their inverse: 231, 312.
		

Links

• 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.

Crossrefs

Cf. A113247.

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 A113247