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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. 105
Sometimes an alternative form of f(x, u) is easier to use.
PROPOSITION 3.32
We have
f(x, u)=
n0
(x)
n
u
n
n!
.
PROOF Immediate from Corollary 3.29.
The nice, compact form of the double generating function f(x, u)results
in a plethora of recurrence relations on the numbers s(n, k). These recur-
sive formulae are sometimes classified into three sets, triangular, vertical,and
horizontal recurrences. These names are based on the arrays of the numbers
s(n, k)involved.
A triangular recurrence is readily obtained from Lemma 3.19. It is
s(n, k)=s(n 1,k 1) (n 1)s(n 1,k), (3.4)
where n 1.
Now we are go
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Publisher Resources

ISBN: 9781439850527