
Do Not Look Just Yet. Solutions to Odd-Numbered Exercises. 425
9. Theorem 7.5, with its notations, shows that necessarily a
1
= k and
a
1
+ a
2
+ ···+ a
r
= r · k. Thus necessarily a
1
= a
2
= ··· = a
r
= k
and a
r+1
<k, otherwise there would be r + 1 increasing subsequences of
length k that are disjoint. This means that the size of the last column
is m
k
= r. Applying (7.1) with k variables instead of k − 1 and fixing
m
k
= r we get the proof exactly as we got the proof of Theorem 7.4.
11. In any SYT of shape F ,theentryn has to be in one of the inner corners.
So removing n from any such tableaux, we get an SYT of shape F
for
some F
that is part of the summation.
13. N