
250 Combinatorics of Permutations, Second Edition
RR
K
FIGURE 6.11
How corners cause non-cancelling terms.
PROOF We can express both f
F
and f
F
by the hooklength formula.
After that, m! cancels, and so do all hooklengths that belong to hooks not
being in the same row or column as
¯
K.
What happens with the remaining terms? Let
¯
K =(x, y). Looking at the
row of
¯
K, we see that the hook H
x,j
of F is as long as the hook H
x,j+1
of F
,
unless the vertical parts of these hooks are different, that is, unless there is a
corner R at the end of column j of F . In that case, instead of a cancellation, we
get a d(
¯
K,R)/d(
¯
K,
¯
R) factor. An analogous argument applies for the ...