Let $\mathbf V$ be a representation of the symmetric group $S_n$, say irreducible. Pieri's formula describes the induced representation $$ \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}} \mathbf V \boxtimes \mathbf 1 $$ in terms of Young diagrams: it decomposes as a direct sum of representations obtained by adding $k$ boxes to the Young diagram of $\mathbf V$, no two in the same column. On the other hand one can also write $$ \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}}\mathbf V \boxtimes \mathbf 1 \cong \bigoplus_{\binom{n+k}n} \mathbf V.$$ To make this canonical one needs to choose, for each $n$-element subset $S$ of $[n+k]$, a permutation $\pi \in S_{n+k}$ carrying $\{1,\ldots,n\}$ to $S$. We could say that we choose the lexicographically first such permutation, for instance, or anything else that's more convenient.

Can one make Pieri's formula explicit in this latter formula, by directly writing down a decomposition of the right hand side into irreducible representations?