A277080 - OEIS

%I #32 Oct 03 2016 16:05:53

%S 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,

%T 792,0,495,0,220,0,66,0,12,0,1,1,0,60,0,1770,0,34220,0,487635,0,

%U 5461512,0,50063860,0,386206920,0,2558620845,0,14783142660,0,75394027566

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

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

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

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

%e 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.

%e 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.

%e The triangle starts:

%e 1, 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, 0, 3, 0, 3, 0, 1;

%o (Sage) def T(n,k):

%o     if k % 2 == 1:

%o         return 0

%o     return binomial( factorial(n)/2, k/2 )

%Y Row lengths give A038507.

%K nonn,tabf

%O 0,10

%A _Christian Bean_, Sep 28 2016