A277080 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by reverse.

1, 1, 1, 1, 1, 0, 1, 1, 0, 3, 0, 3, 0, 1, 1, 0, 12, 0, 66, 0, 220, 0, 495, 0, 792, 0, 924, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1, 1, 0, 60, 0, 1770, 0, 34220, 0, 487635, 0, 5461512, 0, 50063860, 0, 386206920, 0, 2558620845, 0, 14783142660, 0, 75394027566

Offset: 0

Author

Christian Bean, Sep 28 2016

Keywords

• nonn
• tabf

Comments

The reverse of a permutation is the reverse in one line notation. For example the reverse of 43521 is 12534.

T(n,k) is the number of size k bases of S_n which remain unchanged by reverse.

Examples

			For n = 4 and k = 2 the subsets that remain unchanged by reverse are {4321, 1234}, {1243, 3421}, {4231, 1324}, {1342, 2431}, {1423, 3241}, {1432, 2341}, {2134, 4312}, {3412, 2143}, {2314, 4132}, {3142, 2413}, {4213, 3124} and {4123, 3214} so T(4,2) = 12.
For n = 3 and k = 4 the subsets that remain unchanged by reverse are {231, 321, 132, 123}, {321, 213, 312, 123} and {231, 132, 312, 213} so T(3,4) = 3.
The triangle starts:
1, 1;
1, 1;
1, 0, 1;
1, 0, 3, 0, 3, 0, 1;
		

Crossrefs

Row lengths give A038507.

Programs

Sage

def T(n,k):
    if k % 2 == 1:
        return 0
    return binomial( factorial(n)/2, k/2 )

Formula

T(n,k) = C(n!/2, k/2) if k is even and T(n,k) = 0 if k is odd.