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

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

Author

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

Keywords

• nonn
• tabf

Comments

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

Examples

			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;
  ...
		

Links

• 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

Crossrefs

Row sums give A027851.

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

Formula

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).

Extensions

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