

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, 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
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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), k2*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



