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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
58 Combinatorics of Permutations, Second Edition
PROPOSITION 2.9
The ordinary generating function of the numbers p(n) is
n0
p(n)x
n
=
i=1
1
1 x
i
. (2.2)
PROOF We can decompose the right-hand side as
(1 + x + x
2
+ ···)(1 + x
2
+ x
4
+ ···) ···(1 + x
i
+ x
2i
+ ···).
It is now clear that the coefficient of x
n
in this product is equal to the
number of vectors (c
1
,c
2
, ···) with nonnegative integer coefficients for which
i=1
ic
i
= n. Note that such a vector can have only a finite number of
nonzero coordinates. Finally, there is a natural bijection between these vec-
tors and the partitions of n. This bijection maps (c
1
,c
2
, ···) into the partition
that has c
i
parts equal to ...
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Publisher Resources

ISBN: 9781439850527