is a subset $M$ of $B$ such that for all $(a_1,b_1),(a_2,b_2)\in M$ we have $a_1\neq a_2$ and $b_1\neq b_2$.

Equivalently, a matching is a placement of non-attacking rooks on $B$. (A placement of non-attacking rooks is a placement of rooks such that no two rooks are in the same row or column.) The total number of ways to place non-attacking rooks is called the rook number. There is extensive literature on rook numbers, for example see [@Rooknumber]. We will use the language of rook placements in some of the proofs.

Matchings in a Young diagram $B$ correspond to a matchings in the bipartite graph associated with $B$ by equating the box $(i,j)\in B$ with the edge $x_iy_j$ in the associated bipartite graph.

Let the set of $k$-matchings in a Young diagram $B$ be denoted ${\mathcal{M}}_k(B)$ and let ${\mathcal{M}}(B) = \bigcup_{k\geq 0} {\mathcal{M}}_k(B)$. Also define $m_k(B)= \abs{{\mathcal{M}}_k(B)}$ and $m(B) = \abs{{\mathcal{M}}(B)}$.

Figure \[YD\] is an example of a Young diagram in a $4\times 6$ frame and Figure 2 is a matching of size 3 in that Young diagram. In all our diagrams we label rows and columns using matrix numbering.

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(0,2) grid (6,3); (0,1) grid (5,2); (0,0) grid (3,1); (0,-1) 