
DoNotLookJustYet. Solutionsto
Odd-Numbered Exercises.
Solutions for Chapter 1
1. We have
α(S)=
n
s
1
n − s
1
s
2
− s
1
n − s
2
s
3
− s
2
···
n − s
k
n − s
k
=
n!
s
1
!(n − s
1
)!
·
(n − s
1
)!
(s
2
− s
1
)!(n − s
2
)!
···
(n − s
k
)!
(n − s
k
)!
=
n!
s
1
! · (s
2
− s
1
)! ·····(n − s
k
)!
=
n
s
1
,s
2
− s
1
, ···,n− s
n
.
3. If p = p
1
p
2
···p
n
has k descents, then its complement p
c
clearly has
n − 1 − k descents. Here p
c
is the n-permutation defined by (p
c
)
i
=
n +1− p
i
.
5. Look at the sequence {b
i
}
i
where b
i
= a
i
/a
i−1
.Then{a
i
}
i
is log-
concave if and only if {b
i
}
i
is weakly decreasing, while {a
i
}
i
is unimodal
if and only if once {b
i
}
i
gets to a number that is not larger than 1, it
never grows back above 1. As this second ...