2023/08/08 by Shane Chern, Chern, Shane, James A. Sellers +1 · 2 citations
Mathematics · #05A17 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2308.04348
openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size n by the function PDO(n). Since then, numerous authors have proven a variety of divisibility properties satisfied by PDO(n). Recently, the second author proved the following internal congruences satisfied by PDO(n): For all n≥ 0, PDO(4n) amp;≡ PDO(n) \pmod4,
PDO(16n) amp;≡ PDO(4n) \pmod8. In this work, we significantly extend these internal congruence results by proving the following new infinite family of congruences: For all k≥ 0 and all n≥ 0, PDO(22k+3n) ≡ PDO(22k+1n) \pmod22k+3. We utilize several classical tools to prove this family, including generating function dissections via the unitizing operator of degree two, various modular relations and recurrences involving a Hauptmodul on the classical modular curve X0(6), and an induction argument which provides the final step in proving the necessary divisibilities. It is notable that the construction of each 2-dissection slice of our generating function bears an entirely different nature to those studied in the past literature.