On the coefficients of the characteristic polynomial of a matrix (Q1322277)

As is well known, the coefficients \(p_ i\) \((i=0,1, \dots, n)\) of the characteristic polynomial of a square matrix \(A\) of order \(n\) can be obtained from traces \(q_ k=\text{tr} A^ k\) \((k=0,1, \dots,n-1)\) by Newton's formulas in a recursive way. The author develops an explicit combinatorial formula yielding \(p_ i\) solely as functions of \(q_ k\). As a consequence some combinational identities as well as a statement on the characteristic polynomial of a digraph without even cycles are obtained.