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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
Do Not Look Just Yet. Solutions to Odd-Numbered Exercises. 395
Now let us divide both sides by [n 1]! to obtain
[n]=[k]q
nk
+[n k],
which follows directly from the definition of [m].
23. This can be proved by repeated applications of the result of Exercise 21,
but we prefer a combinatorial argument. The left-hand side provides
the generating function for all k-subsets of [m] according to their subset
sums.
A typical term of the right-hand side is of the form q
mkj+1
mj
k1
,
where j [m k]. We claim that this term will provide the generating
function for those k-subsets of [m] whose largest element is equal to
m j + 1. Indeed, the rest of such a subset is a (k 1)-subset of the
set [m j]. The term q
mjk+1
corrects the shift in the definitio
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Publisher Resources

ISBN: 9781439850527