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<k<=floor(n/2).

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

2,2

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

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

Sequence in context: A340620 A184137 A135610 * A203330 A197442 A113497

Adjacent sequences: A157015 A157016 A157017 * A157019 A157020 A157021

Keyword

easy,nonn,tabf

Author

Allan L. Edmonds and Vladeta Jovovic, Feb 21 2009

Status

approved