Recurrence and congruences for the smallest parts function

Let $\text{spt}(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\text{spt}$ 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 ${\sum }_{n\ge 0}\text{spt}(\mathrm{\ell }n-{\delta }_{\mathrm{\ell }})q^n(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell })$, ${\delta }_{\mathrm{\ell }}:=({\mathrm{\ell }}^2-1)\mathrm{/}24$, by Hecke traces for weight $\mathrm{\ell }+1$ cusp forms on ${\text{SL}}_2(\mathbb{Z})$. We further establish an incongruence result for the $\text{spt}$ function.