A277085 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 90 degrees.

1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 4, 6, 10, 14, 20, 26, 31, 36, 40, 44, 44, 44, 40, 36, 31, 26, 20, 14, 10, 6, 4, 2, 1, 1, 2, 4, 6, 34, 62, 116, 170, 547, 924, 1624, 2324, 5572, 8820, 14616, 20412, 40509, 60606, 95004, 129402, 224406, 319410

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 90 degrees then you get a permutation back.

T(n,k) is the number of size k subsets that remain unchanged by a rotation of 90 degrees.

Examples

			For n = 4 and k = 2, the subsets unchanged by a 90-degree rotation are {4321,1234}, {4231,1324}, {3412,2143} and {3142,2413}. Hence T(4,2) = 4.
Triangle starts:
1, 1;
1, 1;
1, 0, 1;
1, 0, 1, 0, 1, 0, 1;
		

Crossrefs

Row lengths give A038507.

Cf. A037223, A037224.

Formula

T(n,k) = Sum_( C( R(n) - T(n), i ) * Sum_(C(n! - R(n), j) * C(T(n), k - 4*i - 2*j) for j in [0..floor((k-4*i)/2)] for i in [0..floor(k/4)] ) where R(n) = A037223(n) and T(n) = A037224(n).