OFFSET
2,2
COMMENTS
T(n,k) is also the number of involutions (unary operators) on S_n, i.e., endomorphisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - Stanislav Sykora, Nov 03 2016
LINKS
Stanislav Sykora, Table of n, a(n) for n = 2..2501
Wikipedia, Involution (mathematics).
FORMULA
E.g.f.: y*x^2*exp(x)/(2-y*x^2).
T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).
EXAMPLE
For n = 4 we have 12 lists: 6 1-lists: [{1,2}], [{1,3}], [{1,4}], [{2,3}], [{2,4}], [{3,4}] and 6 2-lists: [{1,2},{3,4}], [{3,4},{1,2}], [{1,3},{2,4}], [{2,4},{1,3}], [{1,4},{2,3}] and [{2,3},{1,4}].
MATHEMATICA
Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 04 2016 *)
PROG
(PARI) nmax=100; a=vector(floor(nmax^2/4)); idx=0;
for(n=2, nmax, for(k=1, n\2, a[idx++]=n!/(2^k*(n-2*k)!)));
a \\ Stanislav Sykora, Nov 03 2016
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Allan L. Edmonds and Vladeta Jovovic, Feb 21 2009
STATUS
approved