A058131 -id:A058131 - cp's OEIS frontend

A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

1, 1, 1, 1, 3, 1, 2, 9, 6, 2, 1, 26, 30, 16, 5, 1, 98, 142, 111, 54, 15, 1, 455, 718, 713, 482, 215, 53, 3, 4018, 4277, 4637, 3919, 2414, 996, 222, 2, 101910, 36124, 32937, 31308, 24047, 13758, 5294, 1078, 1, 10054303, 708355, 290278, 260758, 229411, 164216, 88178, 31867, 5994

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  2,  9,   6,   2;
  1, 26,  30,  16, 5;
  1, 98, 142, 111, 54, 15;
  ...

Index entries for sequences related to monoids

Row sums give A058131.
Main diagonal: A006966.
Columns 1..2: A000688, A058143.
Cf. A058116, A058137, A058150, A058159, A058160.

a(30)-a(55) from Andrew Howroyd, Feb 15 2022

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 72, 12151, 53146457

Offset: 0

Jianing Song, Oct 26 2019

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

a(n) = (A057771(n)+A057996(n))/2.

cp's OEIS Frontend

A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

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A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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