Counting numerical semigroups by genus and some cases of a question of Wilf
Counting numerical semigroups by genus and some cases of a question of Wilf
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按属数计算数值半群以及 Wilf 问题的一些情况
DOI：
10.1016/j.jpaa.2011.10.038
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发表时间：
2012
影响因子：
0.8
通讯作者：
N. Kaplan
中科院分区：
文献类型：
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作者：
N. Kaplan
一个数值半群的属就是它的补的大小。本文证明了用格计数数值半群的一些结果。2008年Bras-Amorós推测，g+1属的半群数与g属的半群数之比，随着g的增大，趋近于黄金比例φ。虽然最近有几篇论文提供了计数半群的界限，但这个猜想仍然没有得到解决。在本文中，我们将证明一类半群，其两次属小于最小非零元素的三倍，像斐波那契数一样增长，并给出了这个猜想成立的一个可能的原因。我们还将验证1978年的威尔夫问题对这些半群和某些其他情况成立。我们还将证明在某些情况下，我们可以通过只计算最大嵌入维数的半群来计算具有一定格数和多重数的数值半群，并且我们总是可以用单个有理多交体上整数点的数目来解释g属半群的数目。我们还讨论了与Blanco， García-Sánchez和Puerto最近的工作的联系，并提到了几个进一步开放的问题。
The genus of a numerical semigroup is the size of its complement. In this paper, we will prove some results about counting numerical semigroups by genus. In 2008, Bras-Amorós conjectured that the ratio between the number of semigroups of genus g+1 and the number of semigroups of genus g approaches ϕ, the golden ratio, as g gets large. Though several recent papers have provided bounds for counting semigroups, this conjecture is still unsolved. In this paper, we will show that a certain class of semigroups, those for which twice the genus is less than three times the smallest nonzero element, grows like the Fibonacci numbers, suggesting a possible reason for this conjecture to hold. We will also verify that a 1978 question of Wilf holds for these semigroups and in certain other cases. We will also show that in several situations we can count numerical semigroups of certain genus and multiplicity by counting only semigroups of maximal embedding dimension, and that we can always interpret the number of semigroups of genus g in terms of the number of integer points in a single rational polytope. We also discuss connections with recent work of Blanco, García-Sánchez and Puerto, and mention several further open problems.