Linear extensions of finite posets and a conjecture of G. Kreweras on permutations
Let x be a permutation of \([n]=\{1,2,...,n\}\), n an integer \(\geq 2\). The signature of x is the (n-1)-tuple \(\epsilon =\{\epsilon_ 1,\epsilon_ 2,...,\epsilon_{n-1}\}\) where \(\epsilon_ i\) is the sign \(''+''\) if \(x_ i<x_{i+1}\) and ''-'' otherwise. Let \(P_{\epsilon}\) be the set of permutations of [n] with given signature \(\epsilon\). It is proved that the number of solutions \((x,y)\in P_{\epsilon}\times P_{\epsilon}\) of the equation \(xy=z\) for \(z\in P_{\epsilon}\) does not depend on z in \(P_{\epsilon}\).
scientific article; zbMATH DE number 3869332