Reziprozitätsgesetze in der kombinatorischen Zähltheorie (Q1238811)

Let \(S\) and \(P\) be two counting problems. A reciprocity law establishes a duality between \(S\) and \(P\) which permits the solution of one problem if the other is known. This report gives a survey of problems, examples and theorems in this field. Simplest example: \(f(x)=\tbinom {x+n-1} n\) counts the number of monotone words of length \(n\) taken from an \(x\)-set and \(\overline{f}(x) =\tbinom x n\) counts those words which are in addition injective, the reciprocity law being \(\overline{f}(x)=(-1)^nf(-x)\). More difficult example: Let \(m_n(x)\) be the number of \(n\times n\)-matrices over the non-negative integers all of whose rows and columns sum to \(x\), and let \(\overline{m}_n(x)\) be the number of those matrices which have only positive entries. Theorem: \(\overline{m}_n(x)=(-1)^{n-1}m_n(-x)\).