Recurrence and congruences for the smallest parts function

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet series. As a corollary, we give a closed form expression of the power series $\sum_{n\geq 0}\spt(\ell n-\delta_{\ell})q^n\pmod{\ell}$, $\delta_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1$ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function.