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Journals / Turkish Journal of Science and Technology / 2026 / Volume: 21 Issue: 1

Combinatorial Insights into Numerical Semigroups: Rook Placements and Lah Numbers

Sayısal Yarı Gruplara İlişkin Kombinasyonel İçgörüler: Kale Yerleşimleri ve Lah Sayıları

Meral Süer(BATMAN ÜNİVERSİTESİ)0000-0002-5512-4305
Pages
—
Publication Date
2026-03-30
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Our approach introduces a new combinatorial interpretation of numerical semigroup gaps and reveals novel structural links between algebraic and enumerative concepts.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This paper investigates the interplay between numerical semigroups and enumerative combinatorics through the lens of Young diagrams, rook polynomials, and the Lah numbers.
📊
Key Findings & Data:For a given numerical semigroup S, we associate a Young diagram constructed from the gap set of S.
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Conclusion & Impact:Our approach introduces a new combinatorial interpretation of numerical semigroup gaps and reveals novel structural links between algebraic and enumerative concepts.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Lah numbers · numerical semigroups · rook polynomials

Abstract

This paper investigates the interplay between numerical semigroups and enumerative combinatorics through the lens of Young diagrams, rook polynomials, and the Lah numbers. For a given numerical semigroup S, we associate a Young diagram constructed from the gap set of S. We then compute the rook polynomial corresponding to this diagram and analyze its coefficients. It is observed that these coefficients exhibit a strong connection with the Lah numbers, which count ordered partitions. Our approach introduces a new combinatorial interpretation of numerical semigroup gaps and reveals novel structural links between algebraic and enumerative concepts.

Özet

Bu makale, Young diyagramları, kale polinomları ve Lah sayıları merceğinden sayısal yarıgruplar ve sayma kombinasyonları arasındaki etkileşimi araştırır. Belirli bir sayısal yarıgrup S için, S’nin boşluk kümesinden oluşturulmuş bir Young diyagramı ilişkilendiririz. Daha sonra bu diyagrama karşılık gelen kale polinomunu hesaplar ve katsayılarını analiz ederiz. Bu katsayıların, sıralı bölümlemeleri sayan Lah sayılarıyla güçlü bir bağlantı gösterdiği gözlemlenir. Yaklaşımımız, sayısal yarıgrup boşluklarının yeni bir kombinatoryal yorumunu sunar ve cebirsel ve sayma kavramları arasında yeni yapısal bağlantılar ortaya koyar.

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Cite This Article

Süer, M. (2026). Combinatorial Insights into Numerical Semigroups: Rook Placements and Lah Numbers. Turkish Journal of Science and Technology, Volume: 21 Issue: 1. https://doi.org/10.55525/tjst.1724097

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