Power-partible Reduction and Congruences for Schr\"oder Polynomials

In this note, we apply the power-partible reduction to show the following arithmetic properties of large Schr\"oder polynomials $S_n(z)$ and little Schr\"oder polynomials $s_n(z)$: for any odd prime $p$, nonnegative integer $r\in\mathbb{N}$, $\varepsilon\in\{-1,1\}$ and $z\in\mathbb{Z}$ with $\gcd(p,z(z+1))=1$, we have $$\sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k S_k(z)\equiv 1\pmod {p}\quad \text{and} \quad \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k s_k(z)\equiv 0\pmod {p}.$$