Les chemins de Schr\"oder

TL;DR

This paper explores combinatorial properties of r-Schr"oder paths, including diagonal inversions, parking functions, and formulas for paths with fractional slopes, expanding understanding of these generalized paths.

Contribution

It introduces new formulas and analyses for r-Schr"oder paths, particularly for paths with fractional slopes, enhancing combinatorial enumeration methods.

Findings

Derived a formula for Schr"oder paths with a 1/r slope.

Analyzed diagonal inversions in r-Schr"oder paths.

Studied parking functions on r-Schr"oder paths with integer slopes.

Abstract

Apr\`es avoir pos\'e les d\'efinitions n\'ecessaires \`a la compr\'ehension du sujet, nous discuterons de statistique d'inversion diagonale dans les $r$-Schr\"oder, de chemins de stationnement dans les $r$-Schr\"oder \`a pente enti\`ere et nous d\'evelopperons une formule pour les chemins de Schr\"oder ayant une fraction unitaire comme pente. After setting the definitions we discuss of diagonal inversions on $r$-Shr\"oder paths, of parking fonctions on $r$-Shr\"oder paths with an integer slope and we develop a formula for Shr\"oder paths with a $1/r$ slope, with $r$ an integer.