On 3-regular partitions with odd parts distinct

Abstract

We consider $\text {pod}_3(n)$, the number of 3-regular partitions with odd parts distinct, whose generating function is

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$

where

$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$

For each $\alpha >0$, we obtain the generating function for

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$

where $4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}$ if $\alpha$ is even, $4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}$ if $\alpha$ is odd.

We show that the sequence {$\text {pod}_3(n)$} satisfies the internal congruences

$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$

(0.1)

$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$

(0.2)

and

$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$

(0.3)