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Combinatorics of Permutations, 2nd Edition
book

Combinatorics of Permutations, 2nd Edition

by Miklos Bona
April 2016
Intermediate to advanced content levelIntermediate to advanced
478 pages
14h 44m
English
Chapman and Hall/CRC
Content preview from Combinatorics of Permutations, 2nd Edition
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
i1
.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 ...
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Publisher Resources

ISBN: 9781439850527