The permuted analogues of three Catalan sets
Define the odd double factorial \(K_ n\) by \(K_ n=(2n-1)!!=(2n- 1)\cdot(2n-3)\cdot\dots\cdot 3\cdot 1\). The authors discuss three objects which are counted by the Catalan numbers but, if certain order restrictions are removed, are counted by \(K_ n\). These are sequences of paired events, complete binary trees, and arc diagrams of \(n\) chords joining \(2n\) points on a circle. The distributions of certain parameters in both kinds of sets are compared and it is found that those with the same distributions in the Catalan sets do not always have the same distributions in the permuted sets. The authors also consider the problem of extending bijections between Catalan sets to bijections between the permuted sets.
scientific article; zbMATH DE number 168229
scientific article