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

0,16

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.

Links

Table of n, a(n) for n=0..60.

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).

Example

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.

Sequence in context: A082379 A167379 A213476 * A094589 A071425 A115065

Adjacent sequences: A277082 A277083 A277084 * A277086 A277087 A277088

Keyword

nonn,tabf

Author

Christian Bean, Sep 28 2016

Status

approved