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

0,10

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.

Links

Table of n, a(n) for n=0..59.

Formula

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

Example

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;

Prog

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

Crossrefs

Row lengths give A038507.

Sequence in context: A030220 A219240 A349612 * A055240 A174559 A334040

Adjacent sequences: A277077 A277078 A277079 * A277081 A277082 A277083

Keyword

nonn,tabf

Author

Christian Bean, Sep 28 2016

Status

approved