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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
In Many Circles. Permutations as Products of Cycles. 95
s
s
s
s
s
T
s
T
T
T
T
T
T
T
2
6
5
4
9
3
7
8
s
1
2
3
4
6
7
8
9
5
s
FIGURE 3.3
Labeling the edges of T .
construct a bijection h : T
n
C
n
,whereC
n
is the set of all (n 1)-tuples
(s
n
,s
n1
, ···,s
2
) of transpositions satisfying s
n
s
n1
s
n2
···s
2
=(12···n).
Let T T
n
. Then for every vertex i ∈{2, 3, ···,n}, there is exactly one
path from 1 to i in T .If(a, i) is the last edge in this path, then we label the
edge (a, i)bys
T
i
. See Figure 3.3 for an example.
Let s
i
denote the transposition interchanging the two endpoints of s
i
.Set
C
T
= s
n
T
s
n1
T
···s
2
T
;thenC
T
is a cyclic permutation because of the previ-
ous lemma. As an intermediate step in
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Publisher Resources

ISBN: 9781439850527