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

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, A023815.

Cf. A383885, A383886.

Formula

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)).