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

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 \(13\).