A023815 -id:A023815 - cp's OEIS frontend

A383871 Number of labeled 3-nilpotent semigroups of order n.

0, 0, 6, 180, 11720, 3089250, 5944080072, 147348275209800, 38430603831264883632, 90116197775746464859791750, 2118031078806486819496589635743440, 966490887282837500134221233339527160717340, 17165261053166610940029331024343115375665769316911576, 6444206974822296283920298148689544172139277283018112679406098010

Offset: 1

Elijah Beregovsky, May 13 2025

nonn

A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
If r is the smallest such number, then S is said to have nilpotency degree r.
This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.

H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.

Andreas Distler and James D. Mitchell, The number of nilpotent semigroups of degree 3, arXiv:1201.3529 [math.CO], 2012.
Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, Semirigidity and the enumeration of nilpotent semigroups of index three, arXiv:2411.00466 [math.CO], 2024.
Pierre A. Grillet, Counting Semigroups, Communications in Algebra, 43(2), 574-596, (2014).
D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc. 55 (1976), 227-232.
Index entries for sequences related to semigroups

Cf. A023814, A023815.

Cf. A383885, A383886.

a(n) = Sum_{2 <= m <= b(n)} binomial(n,m) * m * Sum_{0 <= i <= m-1} (-1)^i * binomial(m-1,i) * (m-i)^((n-m)^2), where b(n) = floor(n + 1/2 - sqrt(n-3/4)).

A346414 Number of labeled totally ordered commutative semigroups.

Original entry on oeis.org

1, 1, 4, 20, 114, 710, 4726, 33157, 243048, 1850817, 14590692

Offset: 0

Peter Jipsen, Jul 15 2021

Petr Gajdoš and Martin Kuřil, Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10, Semigroup Forum, 89 (2014), 639-663.

Cf. A023815.

A118100 Number of commutative semigroups of order <= n.

Original entry on oeis.org

1, 2, 5, 17, 75, 400, 2543, 19834, 241639, 11787482, 3530717819

Offset: 0

Jonathan Vos Post, May 11 2006

A001426(n) is the number of commutative semigroups of order n. A001426(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n). 2, 5, 17, 2543 and 241639 are primes.

			a(8) = 1 + 1 + 3 + 12 + 58 + 325 + 2143 + 17291 + 221805 = 241639.

Index entries for sequences related to semigroups.

Cf. A001423, A001426, A023815, A027851, A058105, A058116.

a(n) = Sum_{i=1..n} A001426(i).

a(9)-a(10) added using the terms in A001426 by Miles Englezou, May 27 2025

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A383871 Number of labeled 3-nilpotent semigroups of order n.

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A346414 Number of labeled totally ordered commutative semigroups.

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A118100 Number of commutative semigroups of order <= n.

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