Covering relation in the poset of partitions of an integer

Statement

Let $t$ be a positive integer. Let $\mathcal{P}(t)$ be the poset of partitions of $t$ under the dominance order. A partition is written as the list of its parts.

Let $V=(v_1,\dots,v_m)$ and $W=(w_1,\dots,w_n)$ be partitions of $t$, and suppose $V$ covers $W$ in $\mathcal{P}(t)$.

Then $n\le m+1$, and there are integers $i<j$ with $1\le i<j\le n$ such that all of the following hold.

1. $v_\alpha=w_\alpha$ for every $\alpha<i$.
2. $v_\beta=w_\beta$ for every $\beta$ with $j<\beta\le m$.
3. $v_i=w_i+1$.
4. One of two cases occurs. Either $j\le m$ and $w_j=v_j+1$, or $j=n=m+1$ and $w_j=1$.
5. If $j>i+1$, then $v_\gamma=w_\gamma=v_i-1$ for every $\gamma$ with $i<\gamma<j$.