|
%I #7 May 02 2016 10:35:28
%S 0,0,2,8,56,336,2496,19968,181248,1812480,19956480,239477760,
%T 3113487360,43588823040,653836861440,10461389783040,177843708887040,
%U 3201186759966720,60822550111518720,1216451002230374400
%N 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.
%C 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)
%H H. Barcelo, B. Sagan and S. Sundaram, <a href="http://www.math.msu.edu/~sagan/Papers/cpc.pdf">Counting permutations by congruence class of major index</a>, preprint, 2005.
%H H. Barcelo, B. Sagan and S. Sundaram, <a href="http://dx.doi.org/10.1016/j.aam.2006.03.002">Counting permutations by congruence class of major index</a>, Advances in Applied Mathematics, Volume 39, Issue 2, August 2007, Pages 269-281.
%F 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
%e 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.
%Y Cf. A113247.
%K nonn
%O 0,3
%A _Bruce E. Sagan_, Oct 20 2005
|
