A383885 Number of nonisomorphic 3-nilpotent semigroups of order n.

0, 0, 1, 9, 118, 4671, 1199989, 3661522792, 105931872028455, 24834563582168716305, 53061406576514239124327751, 2017720196187069550262596208732035, 2756576827989210680367439732667802738773384, 73919858836708511517426763179873538289329852786253510, 29599937964452484359589007277447538854227891149791717673581110642

Offset: 1

Author

Elijah Beregovsky, May 13 2025

Keywords

• nonn

Comments

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.

References

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

Links

• 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

Crossrefs

Cf. A023814, A027851.

Cf. A383871, A383886.

Formula

a(n) = A383871(n)/n! * (1+o(1)). See Grillet paper in Links.

For exact formula see the Distler and Mitchell paper.