Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice

Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\map t n$ denote the number of ways $n$ can be partitioned into parts which are specifically not multiples of $3$.

Let $\map v n$ denote the number of ways $n$ can be partitioned such that no part appears twice.

Then:

$\forall n \in \Z_{>0}: \map t n = \map v n$

Proof

This theorem requires a proof. In particular: Chapter $12$ of 1971: George E. Andrews: Number Theory You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Exercise $14$