Multinomial Sum Formulas of Multiple Zeta Values

For a pair of positive integers $n,k$ with $n\ge 2$, in this paper we prove that $$\sum _{r=1}^k\sum _{\mathrm{\mid }\mathbf{\alpha }\mathbf{\mid }=\mathbf{k}}\left(\genfrac{}{}{0px}{}{k}{\mathbf{\alpha }}\right)\zeta (n\mathbf{\alpha })=\mathbf{\zeta }(\mathbf{n}{)}^{\mathbf{k}}=\sum _{\mathbf{r}=\mathbf{1}}^{\mathbf{k}}\sum _{\mathbf{\mid }\mathbf{\alpha }\mathbf{\mid }=\mathbf{k}}\left(\genfrac{}{}{0px}{}{\mathbf{k}}{\mathbf{\alpha }}\right)(-\mathbf{1}{)}^{\mathbf{k}-\mathbf{r}}{\mathbf{\zeta }}^{\star }(\mathbf{n}\mathbf{\alpha }),$$ where $\mathbf{\alpha }=({\mathbf{\alpha }}_{\mathbf{1}},{\mathbf{\alpha }}_{\mathbf{2}},\dots ,{\mathbf{\alpha }}_{\mathbf{r}})$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$\sum _{r=1}^{2k}r!\left\{\genfrac{}{}{0px}{}{2k}{r}\right\}=\sum _{p=1}^k\sum _{q=1}^k\left\{\genfrac{}{}{0px}{}{k}{p}\right\}\left\{\genfrac{}{}{0px}{}{k}{q}\right\}p!q!D(p,q),$$ where $\left\{\genfrac{}{}{0px}{}{k}{p}\right\}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.