Young Diagram Decompositions for Almost Symmetric Numerical Semigroups

This paper introduces new structural decompositions for almost symmetric numerical semigroups through the combinatorial lens of Young diagrams. To do that, we use the foundational correspondence between numerical sets and Young diagrams, which enables a visual and algorithmic approach to studying properties of numerical semigroups. Central to the paper, a decomposition theorem for almost symmetric numerical semigroups is proved, which reveals that such semigroups can be uniquely expressed as a combination of a numerical semigroup, its dual and an ordinary numerical semigroup.

💡 Research Summary

The paper investigates almost symmetric numerical semigroups through the combinatorial framework of Young diagrams. After recalling basic notions—gaps, Frobenius number, pseudo‑Frobenius numbers, genus, and type—it emphasizes the well‑known inequality 2 g ≥ F + t, with equality characterizing almost symmetric semigroups. The authors then revisit the bijection between proper numerical sets and Young diagrams introduced by Keith and Nath, providing explicit algorithms to translate a set of small elements into a partition (Algorithm 1), to reconstruct the set from a partition (Algorithm 2), and to compute hook lengths for each row (Algorithm 3).

Three diagrammatic sum operations are defined: the bonded sum (⊞B), which glues two diagrams vertically with a one‑row overlap; the end‑to‑end sum (⊞E), which aligns the top‑right corner of the first diagram with the bottom‑left corner of the second; and the conjoin sum (⊞C), which places the leftmost column of the second diagram directly above the rightmost column of the first, reducing the total column count by one. The paper proves that each operation corresponds to a natural operation on the associated numerical sets and that the resulting sets remain proper.

The central contribution is Lemma 5.2 (a symmetric set construction) and Theorem 5.8, the decomposition theorem. The theorem states that any almost symmetric numerical semigroup S whose pseudo‑Frobenius numbers are consecutive except for the Frobenius number itself can be uniquely expressed as
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