threshold bipartite graphs to rook placements in the Young diagram of the partition. We will make use of this connection heavily in the proof of Theorem \[bipartite\].

Given $(\ell,r,e)$, threshold bipartite graphs $G$ with vertex classes of size $\ell$ and $r$ and $\abs{E(G)} = e$ are in bijective correspondence with partitions of the integer $e$ with at most $\ell$ parts each of size at most $r$. From a partition $\lambda$ we can construct the associated bipartite graph $G_{\lambda}$ by letting $E(G_{\lambda}) = \{x_iy_j: j\leq \lambda_i\}$. Given a threshold bipartite graph we get a partition of $\abs{E(G)}$ by letting the degree of each vertex on the left be the size of a part. We will use this correspondence to represent threshold bipartite graphs as Young diagrams.

Let $B$ be a subset of $[\ell]\times [r]$. If $(i,j)\in B$ we call $(i,j)$ a *box* of $B$. We call $B$ a *Young diagram* if for all $(i,j)\in B$ with $i>1$ we have $(i-1,j)\in B$ and for all $(i,j)\in B$ with $j>1$ we have $(i,j-1)\in B$. We call $[\ell]
\times [r]$ the *frame* of the Young diagram and we’ll say that the Young diagram has dimensions $\ell\times r$. A *matching* in a Young diagram $B$ 