Multiplicative congruences for Andrews's even parts below odd parts function and related infinite products
Frank Garvan, Connor Morrow
TL;DR
This work proves multiplicative congruences for Andrews's even parts below odd parts function $\overline{\mathcal{EO}}(n)$ and related eta-quotients by deploying Fricke involutions and Newman's trace method for half-integer weight eta-quotients. Central to the approach is expressing generating functions as eta-quotients, constructing trace-based modular objects $G(\tau)$ and $G^*(\tau)$, and deriving explicit coefficient identities that yield congruences modulo large powers of two, including a primary result for $\alpha(n)$ with primes $\ell$ and offsets $N_{\ell}$. The authors extend the framework via Faber polynomials and Atkin–Ono-type machinery to broader eta-quotients, obtaining eigenvalue congruences $\lambda_{r,s,p,\ell}$ and new $2$-power congruences for the overpartition function $\overline{p}(n)$. These results illuminate multiplicative structures in eta-quotient congruences, connect to Ramanujan’s mock theta functions, and suggest avenues for generalizing to non-genus-zero cases and deeper arithmetic phenomena. The work thus broadens the scope of modular-congruence techniques in partition theory with concrete, computable congruences and explicit constants.
Abstract
We prove multiplicative congruences mod $2^{12}$ for George Andrews's partition function, $\overline{\mathcal{EO}}(n)$, the number of partitions of $n$ in which every even part is less than each odd part and only the largest even part occurs an odd number of times. We find analogous congruences for more general infinite products. These congruences are obtained using Fricke involutions and Newman's approach to half integer weight Hecke operators on eta quotients, and were inspired by Atkin's multiplicative congruences for the partition function.