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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
78 Combinatorics of Permutations, Second Edition
28. Let n<k
n
2
.Provethat
b(n +1,k)=b(n +1,k 1) + b(n, k) b(n, k n 1).
29. (+) Find a formula for
n
k=0
(1)
k
n
k
.
30. (+) Consider the following refinement of the Eulerian polynomials. Let
A
n,k
(q)=
p
q
maj(p)
,
where the sum is taken over all n-permutations having k 1 descents.
These polynomials are often called the q-Eulerian polynomials. Prove
that
[x]
n
=
n
k=1
A
n,k
(q)
x + n k
n
.
31. Let p be a permutation of length n 1. Insert the entry n into all p
in all possible ways. This yields n distinct permutations of length n.
Compute the major index of each of these permutations. Prove that all
these n major indices will be distinct, and that their set will be the set
of integers in the interval [maj(p
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Publisher Resources

ISBN: 9781439850527