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

0,6

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.

Links

Table of n, a(n) for n=0..51.

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).

Example

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.

Sequence in context: A054482 A208234 A092543 * A216222 A090282 A022910

Adjacent sequences: A277080 A277081 A277082 * A277084 A277085 A277086

Keyword

nonn,tabf

Author

Christian Bean, Sep 28 2016

Status

approved