Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for $pod(n)$

Source abstract

The two-block odd partition function $a(n)$ is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the $2$-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation $a(2^km)=a(m)+(2^{k-1}-1)σ(m)$, where $m$ is odd, $k\ge1$, and $σ(m)$ is the sum-of-divisors function. Combined with a previously established arithmetic formula for $a(n)$, this relation yields congruences along the dyadic progressions $2^k(4n+3)$, as well as the families $a\!\left(2^\alpha3^β(12n+11)\right)\equiv0\pmod3$ and $a\!\left(2^α(18n+15)\right)\equiv0\pmod3$. We also use a factorization of the generating function of $a(n)$ involving $\operatorname{pod}(n)$, the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving $\operatorname{pod}(n)$ are congruent to $a(n)$ and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo $4$. As special cases, we obtain infinite families of weighted congruences modulo $3$, $5$, and $13$.