An elementary proof of a conjecture of Saikia on congruences for t-colored overpartitions

The starting point for this work is the family of functions $$\overline{p}_{-t}(n)$$ which counts the number of t-colored overpartitions of n. In recent years, several infinite families of congruences satisfied by $$\overline{p}_{-t}(n)$$ for specific values of $$t\ge 1$$ have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for $$\overline{p}_{-t}(n)$$ for $$t=5,7,11,13$$ . He also included the following conjecture in that paper: Conjecture: For all $$n\ge 0$$ and primes t, we have $$\begin{aligned} \overline{p}_{-t}(8n+1)\equiv & {} 0 \pmod {2}, \\ \overline{p}_{-t}(8n+2)\equiv & {} 0 \pmod {4}, \\ \overline{p}_{-t}(8n+3)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+4)\equiv & {} 0 \pmod {2}, \\ \overline{p}_{-t}(8n+5)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+6)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+7)\equiv & {} 0 \pmod {32}. \end{aligned}$$ Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia’s conjecture holds for all odd integers t (not necessarily prime).