A new interpretation of Catalan numbers

TL;DR

This paper introduces a new combinatorial interpretation of Catalan numbers through sets related to Kashiwara B(infinity) crystals, revealing novel graph structures called S-graphs that connect to hypercubes and crystal theory.

Contribution

It provides a new description of Catalan sets using labelled hypercubes and decomposes associated graphs into subgraphs with remarkable properties relevant to crystal bases.

Findings

H^t sets are Catalan sets of order t.

Associated graphs G_t decompose into (t-1)! subgraphs called S-graphs.

Number of hypercubes corresponds to Catalan number C(t-1).

Abstract

Towards the study of the Kashiwara B(infinity) crystal, sets H^t of functions were introduced given by equivalence classes of unordered partitions satisfying certain boundary conditions. Here it is shown that H^t is a Catalan set of order t, that is to say the cardinality of H^t is the t-th Catalan number C(t). This is a new description of a Catalan set and moreover admits some remarkable features. Thus to H^t there is an associated labelled graph G_t which is shown to have a canonical decomposition into (t-1)! subgraphs each with 2^{t-1} vertices. These subgraphs, called S-graphs, have some tight properties which are needed for the study of B(infinity). They are described as labelled hypercubes whose edges connecting vertices with equal labels are missing. It is shown that the number of distinct hypercubes so obtained is again a Catalan number, namely C(t-1). They define functions which depend on a coefficient set of non-negative integers. When the latter are non-zero and pairwise distinct, the vertices of the S-graphs describe distinct functions. Moreover this property is retained if certain edges are deleted and certain vertices identified. In particular when these coefficients are all equal and non-zero, it is shown that every hypercube degenerates to a simplex, resulting in exactly t distinct functions, which for example are exactly those needed in the description of B(infinity) in type A.