A combinatorial interpretation of the Catalan transform of the Catalan numbers

TL;DR

This paper provides a combinatorial interpretation of the Catalan transform of Catalan numbers, showing it counts specific functions with particular non-intersecting properties.

Contribution

It introduces a new combinatorial interpretation of the Catalan transform of Catalan numbers, linking it to functions with certain non-intersecting conditions.

Findings

Catalan transform of Catalan numbers counts specific functions.

Provides a simple combinatorial interpretation.

Connects Catalan transform to non-intersecting function properties.

Abstract

The Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it counts functions f:[1,n] -> [1,n] satisfying the condition that, for all i<j, f(j)-(j-i) is not in the interval [1,f(i)-1].