4D Dyck triangle and its projections

TL;DR

This paper explores the multidimensional Dyck triangle and its various plane projections, revealing new perspectives on classic combinatorial structures and their multidimensional generalizations.

Contribution

It introduces the concept of a 4D Dyck triangle and analyzes six 2D and four 3D projections, expanding understanding of Dyck path structures.

Findings

Identification of six 2D projections of the 4D Dyck triangle

Analysis of four 3D projections of the 4D Dyck triangle

Connection between Dyck path parameters and Catalan number decompositions

Abstract

The classic Dyck triangle, the Catalan triangle, and the Catalan convolution matrix are plane projections of the multidimensional Dyck triangle. In the Dyck path, each node is uniquely determined by two of four interrelated parameters: (i) the position of the current parenthesis, (ii) the current unbalance of the parentheses, (iii) the number of viewed left parentheses, and (iv) the same for right parentheses. The last two parameters can be redefined, respectively, as the index of the current Catalan number and the index of the summand in the decomposition of the Catalan number into the sum of squares (Dyck squares). For the 4D Dyck triangle, we consider six 2D projections (some of them are not yet in demand) and four 3D projections.