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Deux propriétés combinatoires des nombres de Schröder. (Two combinatorial properties of Schröder numbers). (French) Zbl 0669.05002

We establish that the Schröder number \(R_ n=\sum^{n}_{i=0}\left( \begin{matrix} 2n-i\\ i\end{matrix} \right)C_{n-i}\) (where \(C_ n\) is the Catalan number \((2n)!/n!(n+1)!)\) satisfies the formula: \(R_{n+1}=2\sum^{\lfloor n/2\rfloor}_{i=0}3^{n-2i}2^ i\left( \begin{matrix} n\\ 2i\end{matrix} \right)C_ i\). As a corollary we prove that \((R_ nR_{n+2}-R_{n+1}R_{n+1})/2\) counts a family of coloured lattice paths. Proofs are purely bijective and use intermediate objects as trees and paths in the plane.

MSC:

05A15 Exact enumeration problems, generating functions
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