Congruences for the Andrews spt function

Ramanujan-type congruences for the Andrews spt(n) partition function have been found for prime moduli $5\leq \ell \leq 37$ in the work of Andrews [Andrews GE, (2008) J Reine Angew Math 624:133–142] and Garvan [Garvan F, (2010) Int J Number Theory 6:1–29]. We exhibit unexpectedly simple congruences for all $\ell \geq 5$ . Confirming a conjecture of Garvan, we show that if $\ell \geq 5$ is prime and $({\textstyle\frac{-\delta}{\ell}})=1$ , then $spt[{\textstyle\frac{\ell ^{2}(\ell n+\delta)+1}{24}}]\equiv 0$ ( ${\rm mod}\ \ell $ ). This congruence gives $(\ell -1)/2$ arithmetic progressions modulo $\ell ^{3}$ which support a mod $\ell $ congruence. This result follows from the surprising fact that the reduction of a certain mock theta function modulo $\ell $ , for every $\ell \geq 5$ , is an eigenform of the Hecke operator $T(\ell ^{2})$ .