MathDB
disjoint cycles

Source: 2011 Romania TST,problem 4

February 4, 2012
linear algebramatrixcombinatorics proposedcombinatorics

Problem Statement

Given an integer n≥2n\ge 2, compute ∑σsgn(σ)nℓ(σ)\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}, where all nn-element permutations are considered, and where ℓ(σ)\ell(\sigma) is the number of disjoint cycles in the standard decomposition of σ\sigma.
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