Number of n-element permutations with exactly k inversions

tytus · 2005-01-31, 14:53

How to find the number of n-element permutations with exactly k inversions?

A pair of indices (i,j), 1<=i<=j<=n, is an inversion of the permutation A if ai>aj.

JuanTutors · 2005-01-31, 19:47

Not to be abnoxious, but is this a homework problem?

tytus · 2005-01-31, 19:55

No, it's not a homework problem... I just would like to know (maybe I should say I have to know) the formula to get the number of n-element permutations with k inversions with given n,k...

geoff · 2005-02-04, 10:10

This is problem 3.20 in "Combinatorial Problems and Exercises" by L. Lovász:

Let a_n,k denote the number of those permutations p of {1, ..., n} which have k inversions (pairs i < j with p(i) > p(j)). Prove that S_k=1 a_n,k x^k = (1+x)(1+x+x²) ... (1+x+...+x^n-1).

(The upper limit of the summation is n choose 2, I don't know how to code that here.)

2005-01-31, 14:53	#1
tytus Jan 2005 2² Posts	Number of n-element permutations with exactly k inversions How to find the number of n-element permutations with exactly k inversions? A pair of indices (i,j), 1<=i<=j<=n, is an inversion of the permutation A if ai>aj.

2005-02-04, 10:10	#4
geoff Mar 2003 New Zealand 10010000101₂ Posts	This is problem 3.20 in "Combinatorial Problems and Exercises" by L. Lovász: Let a_n,k denote the number of those permutations p of {1, ..., n} which have k inversions (pairs i < j with p(i) > p(j)). Prove that S_k=1 a_n,k x^k = (1+x)(1+x+x²) ... (1+x+...+x^n-1). (The upper limit of the summation is n choose 2, I don't know how to code that here.) Last fiddled with by geoff on 2005-02-04 at 10:11

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2005-01-31, 19:47	#2
JuanTutors Mar 2004 21C₁₆ Posts	Not to be abnoxious, but is this a homework problem?

2005-01-31, 19:55	#3
tytus Jan 2005 2² Posts	No, it's not a homework problem... I just would like to know (maybe I should say I have to know) the formula to get the number of n-element permutations with k inversions with given n,k...