Caminos de Dyck contenidos en un diagrama de Ferrers

TL;DR

This paper develops algorithms using Ferrers diagrams to count subsets of Dyck paths contained within them, providing formulas related to Catalan numbers and verified through SAGE coding.

Contribution

It introduces new comparison and decomposition methods for Ferrers diagrams to enumerate Dyck path subsets, implemented as algorithms in SAGE.

Findings

Formulas in terms of Catalan numbers for Dyck path subsets

Algorithms for enumeration verified with SAGE

Enhanced understanding of Dyck paths within Ferrers diagrams

Abstract

We consider the problem of counting subset of Dyck paths contained in a Ferrers diagram. This enumeration concerns to find the number of the elements in a branch of the Kr\'ew\'eras tree. Using the Ferrers diagrams associated with Dyck paths, we have developed methods of comparison and decomposition diagrams to obtain enumerative formulas in terms of Catalan numbers. These methods have been developed in the form of algorithms and SAGE coded for verification. ----- Se considerara el problema de contar el subconjunto de caminos de Dyck contenido en un diagrama de Ferrers. Esta enumeraci\'on ata\~ne al conteo de los elementos en una rama del \'arbol de Kr\'ew\'eras. Mediante el uso de diagramas de Ferrers asociados a los caminos de Dyck, hemos desarrollado m\'etodos de comparaci\'on y de descomposici\'on de diagramas para obtener f\'ormulas enumerativas en t\'erminos de n\'umeros de Catalan. Estos m\'etodos han sido desarrollados en forma de algoritmos y codificados en SAGE para su verificaci\'on.