Linking Congruences for PED and POD partitions

TL;DR

The paper investigates whether congruence multiplicity connects PED and POD partitions in the same way as other partition families. It shows that POD congruences align with overpartitions and PED congruences align with overpartitions having odd parts, rather than suggesting a direct PED–POD equivalence; this is captured through two main results using a conjugated Atkin–Lehner operator $\gamma$ on carefully constructed modular-function families $L_\alpha^{*}$, including explicit relations such as $L^{\mathrm{ped}} \mid \gamma = \tfrac{1}{2} L^{\overline{p_o}} L^{\mathrm{pod}} \mid \gamma = -\tfrac{1}{8} L^{\overline{p}}$ (and the analogous first theorem). The authors derive these relations by computing $P^{*} \mid \gamma$ via eta-transformations, applying the $U_3$-operator, and establishing internal congruences among the same families, grounded in known overpartition congruences and related dissection identities. Collectively, the work illuminates a modular-form pathway linking PED/POD congruences to overpartition congruences, while highlighting that a direct PED–POD logical equivalence via the same method is not currently achieved. The results provide a framework for future modular-form proofs of equivalence and suggest new directions in relating partition families through eta-quotients and Atkin–Lehner symmetries.

Abstract

Recent work of Garvan, Sellers, Smoot, and others has made connections between infinite families of congruences for various partition functions. Here, we apply this approach to families of congruences for PED and POD partitions and find that they are naturally linked to congruence families for overpartitions into odd parts and overpartitions.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Lemmas 6 and 7 of AtkinLehner
• Lemma 2.2
• Lemma 2.3: Theorem 1.64 of OnoWeb
• Lemma 2.4: Theorem 1.65 of OnoWeb
• Lemma 2.5: Chapter 3 of Apostol
• Proposition 5.1
• proof
• Lemma 5.2: Lemma 2.1 of Toh12