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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 One Line and Close. Permutations as Linear Orders. 23
THEOREM 1.32
If a sequence of positive real numbers has real roots only, then it is log-concave.
PROOF Let a
0
,a
1
, ···,a
n
be our sequence, and let P (x)=
n
k=0
a
k
x
k
.
Then for all roots (x, y) of the polynomial Q(x, y)=
n
k=0
a
k
x
k
y
nk
,theratio
(x/y) must be real. (Otherwise x/y would be a non-real root of P (x)). There-
fore, by Rolle’s Theorem, this also holds for the partial derivatives ∂Q/∂x and
∂Q/∂y. Iterating this argument, we see that the polynomial
a+b
Q/∂x
a
∂y
b
also has real zeros, if a + b n 1. In particular, this is true in the special
case when a = j 1, and b = n j 1, for some fixed ...
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Publisher Resources

ISBN: 9781439850527