An inversion of a permutation o is a pair i σj. Let I(n, k) denote the number of permutations in Sn with k inversions. Prove the identity below: ΣI(n, k)x¹ = 1(1+x)(1+x+x²)...(1 + x + ... + x²−¹) k=0
An inversion of a permutation o is a pair i σj. Let I(n, k) denote the number of permutations in Sn with k inversions. Prove the identity below: ΣI(n, k)x¹ = 1(1+x)(1+x+x²)...(1 + x + ... + x²−¹) k=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discrete Math
Explain your steps, Please!
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![i=0
An inversion of a permutation o is a pair i < j such that σį > σj. Let
I(n, k) denote the number of permutations in Sn with k inversions.
Prove the identity below:
n
ΣI(n, k)æk = 1(1 + x)(1 + x + x²)...(1 + x + ... + x²−¹)
k=0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2029095-ed0d-4011-975c-f2065b45a57d%2F7054ab5a-d661-4b5c-9cd2-997db3b19c0d%2Fzqm2f9m_processed.png&w=3840&q=75)
Transcribed Image Text:i=0
An inversion of a permutation o is a pair i < j such that σį > σj. Let
I(n, k) denote the number of permutations in Sn with k inversions.
Prove the identity below:
n
ΣI(n, k)æk = 1(1 + x)(1 + x + x²)...(1 + x + ... + x²−¹)
k=0
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