in-corner. Showing that $B'$ has fewer matchings than $B$ is equivalent to showing that there are fewer matchings in $B^+$ that contain $P'$ and not $P$ than matchings in $B^+$ that contain $P$ and not $P'$. To do this we will define an injection from the collection of $k$-matchings of $B^+$ that contain $P'$ to the collection of $k$-matchings of $B^+$ that contain $P$.

Define $S:={\left\{(a,j)\;:\;i'<a<i\right\}}$ and $T:={\left\{(i',b)\;:\; j<b<j'\right\}}$. The injection will be defined in two parts: firstly on those matchings of $B$ that don’t intersect $S$ and secondly on those that do.

(-1,0) rectangle (10,6);

(-1.5, 1.75) node [$i$]{}; (1.75 , 6.5) node [$j$]{}; (-1.5, 4.25) node [$i'$]{}; (6.75, 6.5) node [$j'$]{};

(2,4) grid (3,4.5); (2,4) – (3,4); (2,4) – (2,4.5); (4,4.25) – (5,4.25); (6,4) grid (7,4.5); (6,4) – (6,4.5); (6,4) – (7,4); (2.5,4.5) – (6.5,4.5) node \[black,midway,yshift=.5cm\][$T$]{}; (6.75,4.25) node [$P'$]{};

(1.5,3.5) grid (2,4); (1.5, 3.5) – (1.5,4); (1.5,3.5) – (2,3.5); (1.75,2.75) – (1.75,3.25); (1.5,1.5) grid (2,2.5); (1.5,1.5) – (1.5,2.5); (1.5,1.5) – (2,1.5); (1.5,2) – (1.5,4) node \[black,midway,xshift=-0.6cm\][$S$]{}; (1.75,1.75) node [$P$]{};

(-1,.5) – (-.5,.5); (-.5,.5) – (-.5,1); (-.5,1) – (0,1); (0.2,1) – (.8,1.2); (1,1.5) – (1.5,1.5); (1,1.25) – (1,1.5);

(2,2) – (3,2); (3,2) – (3,2.5); (3.5,2.7) – (4.7,3.2); (6,3.5) 