A140189 -id:A140189 - cp's OEIS frontend

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

Benoit Jubin, May 12 2008

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.

Cf. A140185.

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).

A140185 Number of connected groupoids (categories all of whose morphisms are invertible) with n morphisms.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 1, 6, 3, 2, 1, 6, 1, 2, 1, 17, 1, 6, 1, 6, 2, 2, 1, 17, 3, 2, 6, 5, 1, 4, 1, 57, 1, 2, 1, 19, 1, 2, 2, 16, 1, 6, 1, 5, 3, 2, 1, 58, 3, 6, 1, 6, 1, 17, 2, 15, 2, 2, 1, 14, 1, 2, 5, 284, 1, 4, 1, 6, 1, 4, 1, 61, 1, 2, 4, 5, 1, 6, 1, 58, 18, 2, 1, 17, 1, 2, 1, 14, 1, 12, 1, 5, 2

Offset: 0

Benoit Jubin, May 12 2008

nonn

a(n) >= A000001(n) (number of groups of order n), with equality if and only if n is squarefree (sequence A005117).

Cf. A140186, A140187, A140188, A140189.

a(0) = 1 and if n>0, a(n) = sum(A000001(n/(k^2)),k^2|n). Indeed, if b(n,k) is the number of connected groupoids with n morphisms and k objects, then the only nonzero values are b(0,0) = 1 and b(m.k^2,k) = A000001(m) with m,k>0.

cp's OEIS Frontend

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.

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