Maschke tells you that $\mathbb C[G] \cong \bigoplus_V V^*\otimes V$, where $V$ runs over the irreps of a finite group $G$. Apply this to $G= S_n$ and take dimensions, and you get $n! = \sum_{\lambda \vdash n} \#SYT(\lambda)^2$, which can be proven combinatorially with the Robinson-Schensted correspondence.
Generalizing this, consider the $GL(k)\times GL(n)$-representation $Sym(Hom(\mathbb C^k,\mathbb C^n)) \cong \bigoplus_{\lambda_1 \geq \ldots \geq \lambda_{\min(k,n)}} (V^{GL(k)}_\lambda)^* \otimes V^{GL(n)}_\lambda$. The LHS has a basis of monomials, corresponding to $k\times n$ matrices of naturals. The RHS has a basis of pairs of same-shape SSYT. This $\cong$ can be proven combinatorially, the RSK correspondence. There is another such story with $Alt$ in place of $Sym$.