a box can be added there to create an out-corner.

\[corners\]

(0,2) grid (6,3); (0,1) grid (5,2); (0,0) grid (3,1); (0,-1) grid (2,0); (0,3) rectangle (6,-2); (5.5,2.5) node [O]{}; (4.5,1.5) node[O]{}; (2.5,.5) node[O]{}; (1.5,-.5) node[O]{}; (5.5, 1.5) node [I]{}; (3.5,.5) node [I]{}; (2.5,-.5) node [I]{}; (.5,-1.5) node [I]{};

In Figure \[corners\] the in-corners are labeled with $I$ and the out-corners are labeled with $O$. Notice that the dimensions of the frame matter; for example, in a $4\times 6$ frame $(5,1)$ would not be an in-corner.

We now define a move that removes a box that is an out-corner and puts a box at an in-corner that is “further out". Let $s(P)$ be the sum of the coordinates of $P$.

\[oneblock\] Let $B$ be a Young diagram in an $m\times n$ frame. Suppose $P$ is an out-corner of $B$ and $P'$ is an in-corner of $B$. If $s(P)<
s(P')$ then $B':=B+P'-P$ is a Young diagram in an $m\times n$ frame and has at most as many $k$-matchings as $B$ for all $k\geq 0$. Moreover, $m(B')<m(B)$.

Suppose $P=(i,j)$ and $P'=(i',j')$. The hypothesis states that $i+j<i'+j'$. Define $B^+ = B+P' = B'+P$. Note that $B^+$ is a Young diagram since $P'$ is an 