A157018 Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0

1, 3, 6, 6, 10, 30, 15, 90, 90, 21, 210, 630, 28, 420, 2520, 2520, 36, 756, 7560, 22680, 45, 1260, 18900, 113400, 113400, 55, 1980, 41580, 415800, 1247400, 66, 2970, 83160, 1247400, 7484400, 7484400, 78, 4290, 154440, 3243240, 32432400, 97297200

Offset: 2

Author

Allan L. Edmonds and Vladeta Jovovic, Feb 21 2009

Keywords

• easy
• nonn
• tabf

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

Examples

			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}].
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 2..2501
• Wikipedia, Involution (mathematics).

Crossrefs

Cf. A000262, A000680, A087214, A100861, A126725 (row sums), A129684.

Programs

Mathematica

Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 04 2016 *)

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

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)!).