2025/03/31 by Guadalupe, Russelle
#05A17 #11P81 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2503.23996
Let b(n) be the number of partition triples π=(π1,π2,π3) of n such that π1 consists of distinct odd parts, and π2 and π3 consist of parts divisible by 4. Utilizing modular forms, Lin obtained the generating functions for b(3n+1) and b(3n+2), which yields the congruence b(3n+2)≡ 0\pmod3 for all n≥ 0. We provide in this note elementary proofs of these generating functions by employing q-series manipulations and dissection formulas. We also establish infinite families of internal congruences modulo 3 for b(n).