A079174 -id:A079174 - cp's OEIS frontend

A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.

0, 5, 3306, 178981764, 2483527537092910, 14325590003318891522247046, 50976900301814584087291487087212542367, 155682086691137947272042502251643461917498831796991599, 541851439802559836957713164869818405872834954135521300809796661279574643

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

Each a(n) is equal to the sum of the elements in row n of A079174.

Christian van den Bosch, Closed binary operations on small sets. [Wayback Machine link]
Index entries for sequences related to groupoids.

Cf. A001329, A027851, A079172, A079174.

a(n) = A001329(n) - A027851(n).

More terms from Christian G. Bower, Nov 26 2003

a(8)-a(9) added using the data at A001329 and A027851 by Amiram Eldar, Jul 19 2025

A079175 Number of isomorphism classes of associative closed binary operations (semigroups) on a set of order n, listed by class size.

Original entry on oeis.org

1, 1, 2, 3, 2, 0, 7, 15, 2, 0, 0, 7, 5, 0, 62, 112, 2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221, 2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, 4, 0, 48, 0, 0, 0, 0, 92, 0, 0, 42, 506, 0, 813, 32, 7397, 19684, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

			Triangle T(n,k) begins:
  1;
  1;
  2, 3;
  2, 0, 7, 15;
  2, 0, 0, 7, 5, 0, 62, 112;
  2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8; row 8 was derived from data given in the Distler-Kelsey reference)
C. van den Bosch, Closed binary operations on small sets
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
Index entries for sequences related to semigroups

Row sums give A027851.

Cf. A023814, A027423 (row lengths), A079171, A079174, A079210.

A079174(n,k) + T(n,k) = A079171(n,k).
T(n, A027423(n)) = A058104(n).
A023814(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079172 Number of non-associative closed binary operations on a set of order n.

Original entry on oeis.org

0, 8, 19570, 4294963804, 298023223876769393, 10314424798490535546154887938, 256923577521058878088611477224227878265543, 6277101735386680763835789423207666416102355296268686994710, 196627050475552913618075908526912116283103450944214766927276968172610579252347

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

Each a(n) is equal to the sum of the products of each element in row n of A079174 and the corresponding element of A079210.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A023814, A079173, A079174.

a(n) = A002489(n) - A023814(n).

More terms from Christian G. Bower, Nov 26 2003

More terms from Jinyuan Wang, Mar 02 2020

cp's OEIS Frontend

A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.

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A079175 Number of isomorphism classes of associative closed binary operations (semigroups) on a set of order n, listed by class size.

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A079172 Number of non-associative closed binary operations on a set of order n.

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