Congruences for two-color partitions with odd smallest part

George E. Andrews; Mohamed El Bachraoui

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arXiv · 2410.14190

Congruences for two-color partitions with odd smallest part

George E. Andrews·Mohamed El Bachraoui

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Abstract

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$.

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BibTeX RIS

George E. Andrews, Mohamed El Bachraoui. 2024-10-18. Congruences for two-color partitions with odd smallest part. https://arxiv.org/abs/2410.14190

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