Action graphs and Catalan numbers
Gerardo Alvarez, Julia E. Bergner, Ruben Lopez

TL;DR
This paper introduces a sequence of directed graphs whose edge counts at each step correspond to Catalan numbers and establishes an isomorphism with planar rooted trees, revealing combinatorial structures.
Contribution
It defines a new class of directed graphs and proves their edge counts match Catalan numbers, linking graph theory with combinatorial tree structures.
Findings
Number of edges added at step k equals the kth Catalan number
Edges at step k are isomorphic to planar rooted trees with k edges
Establishes a combinatorial correspondence between graphs and trees
Abstract
We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step is equal to the th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step and the set of planar rooted trees with edges.
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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications · Analytic Number Theory Research
