Professional paper

Catalan numbers and lattice paths

Ema Dogančić orcid.org/0000-0002-5145-0090 ; studentica, Prirodoslovno-matematički fakultet - Matematički odsjek, Sveučilište u Zagrebu
Ivica Martinjak ; Prirodoslovno-matematički fakultet - Fizički odsjek, Sveučilište u Zagrebu

Abstract

We define Catalan numbers as the sequence of numbers corresponding
to the number of triangulations of a convex polygon. The Catalan
numbers appear in various mathematical contexts and there are many
other combinatorial interpretations of these numbers as well. In this
overview firstly we present basic properties and the Catalan convolution.
We describe fundamental interpretations, which are those for
which the Catalan convolution can be easily seen or there is a simple
correspondence with some of the other interpretations. We enumerate
some notable families of lattice paths. In particular, we show two families
of Dyck paths with constraint on the step (1, −1). Finally, we present
the beautiful Nichols’ bijection between Shapiro and Whitworth
paths.

Keywords

Catalan numbers; lattice paths; Dyck paths; Shapiro paths; Whitworth paths; convolution

Hrčak ID:

212037

URI

https://hrcak.srce.hr/212037

Publication date:

20.7.2018.

Article data in other languages: croatian

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