Witness identities for three Ramanujan congruences

TL;DR

This paper derives explicit witness identities for three Ramanujan congruences related to the partition function, providing concrete formulas that demonstrate these divisibility properties.

Contribution

It introduces explicit witness identities for three specific Ramanujan congruences of the partition function, advancing understanding of their combinatorial structure.

Findings

Explicit witness identities for the three Ramanujan congruences

Demonstration of divisibility properties through concrete formulas

Enhanced understanding of partition function congruences

Abstract

For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.