The combinatorics of sequences that enjoy a curious self-convolutive property

Abstract: In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects called partitions with designated summands. If we restrict our attention to $\mathrm{PDO}(n)$, the number of partitions with designated summands in which all parts are odd, a very curious property emerges. The very unexpected identity $\qquad \sum_{n=0}^\infty \mathrm{PDO}(2n)q^n = \left ( \sum_{n=0}^\infty \mathrm{PDO}(n)q^n \right )^2 $ holds. That is, the sequence $\{\mathrm{PDO}(2n)\}_{n=0}^\infty$ is the convolution of the sequence $\{\mathrm{PDO}(n) \}_{n=0}^\infty$ with itself! Sequences sharing this curious property are now called ``$2$-convolutive,'' and a small handful of such sequences appear in the OEIS. Many authors have called for a combinatorial proof of the $2$-convolutivity of $\mathrm{PDO}(n)$. After a nearly two-year-long collaboration with Chern, Fu, and Sellers, we are happy to announce that we have finally found the requested combinatorial proof. In this talk, we discuss this new proof, along with the combinatorial proofs of the $2$-convolutivity of several other partition functions.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)