×

A sequence of quasipolynomials arising from random numerical semigroups. (English) Zbl 1471.20041

In [J. De Loera et al., Electron. J. Comb. 25, No. 4, Research Paper P4.37, 16 p. (2018; Zbl 1459.20044)], random numerical semigroups are introduced in analogy with the notion of a random graph. The expected number of minimal generators of a random numerical semigroup can be expressed by means of a sequence \(h_{n,i}\), which counts the number of sets with cardinality \(i\) and contained in \([1,n/2) \cap \mathbb{Z}\) that minimally generate an additive semigroup of \(\mathbb{Z}_{\geq 0}\) not containing \(n\).
In this paper, the combinatorics of this sequence is explored. In particular, when the second index is fixed, it is proved that the sequence \(h_{n,i}\) is eventually a quasipolynomial with period \(6\) and its leading coefficient is explicitly described. Moreover, an algorithm to obtain the integers \(h_{n,i}\) is found.

MSC:

20M14 Commutative semigroups

Citations:

Zbl 1459.20044
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] J. Backelin, On the number of semigroups of natural numbers, Math. Scand. 66 (1990), 197-215. · Zbl 0741.11006
[2] J. De Loera, C. O’Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, preprint, 2017,https://arxiv.org/abs/1710. 00979.
[3] J. Rosales and P. Garc´ıa-S´anchez, Numerical Semigroups, Developments in Mathematics, Vol. 20, Springer-Verlag, 2009.
[4] J. Rosales, P. Garc´ıa-S´anchez, J. Garc´ıa-Garc´ıa, J. Jim´enez Madrid, Fundamental gaps in numerical semigroups, J. Pure Appl. Algebra 189 (2004), 301-313.
[5] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,https://oeis.org. 12 · Zbl 1439.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.