A140188 Table read by rows: T(n,k) is the number of groupoids (categories all of whose morphisms are invertible) with n morphisms and k objects.

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 1, 1, 2, 4, 5, 3, 1, 1, 1, 5, 6, 5, 3, 1, 1, 5, 8, 10, 9, 5, 3, 1, 1, 2, 10, 14, 12, 9, 5, 3, 1, 1, 2, 13, 21, 20, 15, 9, 5, 3, 1, 1, 1, 13, 24, 29, 23, 15, 9, 5, 3, 1, 1, 5, 20, 39, 42, 37, 27, 15, 9, 5, 3, 1, 1, 1, 19, 43, 58, 53, 40, 27, 15, 9, 5, 3, 1, 1, 2

Offset: 1

Author

Benoit Jubin, May 12 2008

Keywords

• nonn
• tabl

Comments

The first column is T(n,1) = A000001(n) (number of groups of order n).

T(n,k) >= A136406(n,k).

The sum of the n^th row is A140189(n).

For 2k<=n, T(n,n-k) = A140190(k) does not depend on n.

Crossrefs

Cf. A140185.

Formula

T(n,k) is the sum over the quadratic bi-partitions (n_i,k_i) of (n,k) (see A136406) of the "product" of the A000001(n_i), where the "product" is the usual product except when (n_i1,k_i1)=...=(n_ip,k_ip), in which case a^p is replaced by binomial(a+p-1,p).