Counting the number of isotone selfmappings of crowns

Abstract

Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is$n\left( {\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right) + 2} \right)$. The number of all order-preserving selfmappings of a 2n-element crown is

$$n\left( {2 + \sum\limits_{k = 0}^n {\left[ {\left( {\begin{array}{*{20}c} {n + k} \\ {2k} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {n + k - 1} \\ {2k} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right)} \right]} } \right).$$