Answered: Problem 4: An inversion in a…

Problem 4: An inversion in a permutation T = T1... Tn (written in one-line notation) is a pair of indices 1 pinv(7) (1+x)(1+x+ x²)..· (1 +x + x² +……+x"-1), n-. TES. where x is a variable.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.6: The Algebra Of Matrices

Problem 35E

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Question

Problem 4: An inversion in a permutation T = T1 . .. Tn (written in one-line notation)
is a pair of indices 1 < i <j <n which appear out-of-order when we read a from left
to right. For example, if T =
inv(T) denote the number of inversions in T, so that inv(7) = 4 in our example. Prove
that
23514, the inversions in T are 12,13, 15, and 45. Let
prinv(#) = .. (1+x + x² + ...+ x"-1),
(1+ x)(1+x+ x²)
TESN
where x is a variable.

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