cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

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Author

Benoit Jubin, May 12 2008

Keywords

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).

A140186 Number of connected groupoids (categories all of whose morphisms are invertible) with n more morphisms than objects.

Original entry on oeis.org

2, 1, 2, 2, 1, 2, 3, 5, 2, 2, 2, 5, 2, 2, 3, 15, 1, 5, 2, 5, 3, 2, 3, 15, 3, 2, 6, 4, 2, 4, 7, 51, 1, 4, 3, 14, 1, 2, 4, 14, 1, 6, 4, 4, 3, 3, 6, 52, 2, 5, 2, 7, 1, 15, 4, 13, 3, 2, 2, 13, 4, 2, 18, 267, 1, 4, 3, 5, 1, 9, 7, 50, 2, 2, 4, 4, 2, 6, 8, 52, 15, 2, 3, 15, 1, 2, 3, 14, 1, 10, 3, 5, 4
Offset: 0

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Author

Benoit Jubin, May 12 2008

Keywords

Comments

a(n) >= A000001(n+1) (number of groups of order n+1).

Crossrefs

Cf. A140185, A140187, A140188.

Formula

a(0) = 2 and if n>0, a(n) = sum(A000001((n+k)/(k^2)),k^2|n+k). (see formula for A140185)

A140187 Number of connected groupoids (categories all of whose morphisms are invertible) with n times as many morphisms as objects.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 6, 3, 10, 5, 6, 3, 13, 3, 6, 5, 24, 3, 13, 3, 13, 6, 6, 3, 33, 5, 6, 10, 12, 3, 14, 3, 75, 5, 6, 5, 34, 3, 6, 6, 32, 3, 17, 3, 12, 9, 6, 3, 99, 5, 13, 5, 13, 3, 33, 6, 30, 6, 6, 3, 39, 3, 6, 12, 342, 5, 14, 3, 13, 5, 14, 3, 104, 3, 6, 10, 12, 5, 17, 3, 98, 25, 6, 3, 43, 5, 6, 5, 29
Offset: 0

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Author

Benoit Jubin, May 12 2008, May 16 2008

Keywords

Comments

If n>1, a(n) >= 2 + A000001(n) (number of groups of order n), with equality if and only if n is prime (sequence A000040).

Crossrefs

Cf. A140185, A140186, A140188.

Formula

a(n) = sum(A000001(n/k),k|n). (see formula for A140185)
a(n) = 1 + sum(A000001(n/k),k|n). (the 1 accounts for the empty groupoid; see formula for A140185)

A140189 Number of groupoids (categories all of whose morphisms are invertible) with n morphisms.

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 16, 22, 42, 57, 90, 124, 204, 275, 413, 562, 866, 1161, 1685, 2264, 3308, 4407
Offset: 0

Views

Author

Benoit Jubin, May 12 2008

Keywords

Comments

Row sums of A140188 (see comments and formula there).

Crossrefs

Cf. A140185, A140188.
Showing 1-4 of 4 results.

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