Problem 2

Let \(A=\{1,2,\ldots,n\}\) and \(C\) is the set of all bijective function from \(A\) to itself. Define the function \(T:C\rightarrow \mathbb{R}\) as follow: for all \(f\in C\), \(T(f)\) be the number of pair \((x,y)\in A\times A\) such that \(f(x)>f(y)\) whenever \(x<y\). Evaluate the following summation!
\[\sum\limits_{f\in C}\prod\limits_{x=1}^n (-1)^{T(f)}x^{f(x)}\]