
270 Combinatorics of Permutations, Second Edition
4. Let us keep the notation of the solution of Problem Plus 2. Furthermore,
let us introduce the new notation
e
λ
=
h≥j≥k≥l
a
i
(a
j
− 1)(a
k
− 2)(a
l
− 3)
n(n − 1)(n − 2)(n − 3)
.
It is then proved in [169] that
E(X
i,j
)=c
λ
− d
λ
− d
λ
+ e
λ
+ e
λ
,
where c
λ
is defined in (6.25) below.
5. This probability converges to zero as n goes to infinity as is proved in
[96].
6. This probability converges to 1 as n goes to infinity as is shown in [138].
7. Let the given shape λ have rows of length a
1
,a
2
, ···,a
k
and columns of
length a
1
,a
2
, ···,a
k
. It then follows from a more general result of Peter
H¨ast¨o [169] that
c
λ
= E
d,λ
=
n − 1