数值半群多参数计数的算法与结果
Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees
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中文总结 AI 辅助
研究数值半群多参数计数问题，证明特定条件下的计数公式，提出种子算法的两种改编用于探索相关树结构，并改进算法策略，扩展对序列的认识，计算了多种情况下的半群数量及多参数分解。
中文摘要 AI 辅助
在通过亏格和弗罗贝尼乌斯数来计数数值半群方面已经有很多努力。已知每个亏格的半群数量渐近增长类似于斐波那契数，每个弗罗贝尼乌斯数的半群数量渐近增长方式是每个数是前一个数的前一个数的两倍。我们证明了在重数\(m\)和弗罗贝尼乌斯数\(F\)满足\(m\geq\frac{F + 1}{3}\)的假设下，关于每个弗罗贝尼乌斯数、亏格和重数的半群数量公式。此公式在所需限制下给出了上述增长行为的多参数精确版本。我们还提出了种子算法的两种改编，用于探索给定亏格以下的数值半群的无叶树以及弗罗贝尼乌斯叶判别树，其叶子恰好是给定弗罗贝尼乌斯数的半群。为此，我们改编了递归下降算法，以便在第一种情况下，在没有给定亏格后代的节点处精确修剪树，在第二种情况下，在没有给定弗罗贝尼乌斯数后代的节点处精确修剪树。我们改进了并行化策略，克服了间隙序列和种子序列按位表示中整数长度的先前限制。我们扩展了对三个不同序列的认识。我们得到了\(n_{78}, n_{79}, n_{80}\)，计算了直到128的每个弗罗贝尼乌斯数的半群数量，以及直到128的每个弗罗贝尼乌斯数的不可约数值半群数量。我们还通过前三个跳跃计算了直到亏格80的第一个序列中数字的多参数分解，以及通过重数和亏格计算了直到弗罗贝尼乌斯数128的第二个序列中数字的多参数分解。
英文摘要
A numerical semigroup is a subset of the nonnegative integers, closed under addition and with finite complement. The size of the complement is its genus. The problem of counting semigroups by the largest gap got very important through the so-called Frobenius problem, first documented in 1884. Accordingly, the largest gap is called the Frobenius number. In the last two decades, counting by the genus has become a subject of even more intense study, mostly because of the non-solved conjectures on its monotonic and super-Fibonacci growth. We propose a new approach to counting semigroups by the Frobenius number and by the genus, by introducing two ad-hoc trees. Those are leaf-discriminating trees in the sense that their leaves correspond exactly to the objects we want to count and, so, exploring these trees is optimal. On the theoretical side, it is known that the number of semigroups of each genus grows asymptotically with the genus as the Fibonacci numbers and that the number of semigroups of each Frobenius number grows asymptotically as a two-step doubling sequence. We prove a formula for the number of numerical semigroups of each Frobenius number $F$, genus $g$, and multiplicity $m$ (first nonzero nongap), for $m\geq(F+1)/3$. It is known that asymptotically almost all semigroups satisfy this inequality. This formula gives a multiparameter exact version of the increasing behaviours just mentioned. On the computational side, we implemented a recursive descending algorithm based on the so-called seeds structure, trimming the general semigroup tree exactly at those nodes with no descendants with a given genus, in the first case, or with no descendants with a given Frobenius number, in the second case. We refined the parallelizing strategies and we overcame the previous limitation of the length of integers in the bitwise representation of the gap sequence and the seed sequence.
