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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
400 Combinatorics of Permutations, Second Edition
37. Let A
i
be the set of n-permutations in which i is a fixed point. Then
|A
i
1
∩···∩A
i
k
| =(n k)!, and the result follows by the Principle of
Inclusion–Exclusion.
39. One of these sets is always one larger than the other. For odd n,the
set of n-permutations with exactly one fixed point is larger. For even n,
the set of derangements of length n is larger.
To see this, let G(n)bethenumberofn-permutations with exactly one
fixed point, and note that G(n)=nD
n1
holds for all n 3. Indeed,
to find a permutation counted by G(n), first choose its only fixed point,
then choose a derangement on the remaining n1 entries. On the other
hand, it follows from Corollary 3.57 that D
n
= nD
n1
+(1)
n
.This
proves our claim.
41. The ...
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Publisher Resources

ISBN: 9781439850527