A277083 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 5328, 6728, 7728, 8092, 7728, 6728, 5328, 3823, 2472, 1428, 728, 322, 120, 36, 8, 1, 1, 8, 84, 504, 3178, 15512, 74788, 311144, 1252819, 4577328, 16087512, 52691408, 165911284

Offset: 0

Author

Christian Bean, Sep 28 2016

Keywords

• nonn
• tabf

Comments

A permutation, p, can be thought of as a set of points (i, p(i)). If you plot all the points and rotate the picture by 180 degrees then you get a permutation back.

T(n,k) is the number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

Examples

			For n = 3 and k = 3, the subsets unchanged by rotating 180 degrees are {213,132,123}, {231,312,123}, {321,132,213} and {321,231,312} so T(3,3) = 4.
Triangle starts:
1, 1;
1, 1;
1, 2, 1;
1, 2, 3, 4, 3, 2, 1;
		

Crossrefs

Row lengths give A038507.

Cf. A037223.

Formula

T(n,k) = Sum_( binomial( n! - R(n), i ) * binomial( R(n), k-2*i ) for i in [0..floor(k/2)] ) where R(n) = A037223(n).