Elements of the sets enumerated by super-Catalan numbers

As we know several people tried to get many structures for fine numbers (see [31, Sequence A000957]), while others on Catalan numbers (see [31, Sequence A000108]). Stanley [34,35] gave more than 130 Catalan structures while Deutsch and Shapiro [11] also discovered many settings for the Fine numbers. The structures for Fine numbers and Catalan numbers are intimately related from the list of Fine number occurences in [11], which motivated us to find out more and more super-Catalan structures by the tight link between Catalan numbers and super-Catalan numbers, whose first several terms are 1, 1, 3, 11, 45, 197, · · · (see [31, Sequence A001003]). The purpose of this paper is to give a unified presentation of many new super-Catalan structures. We start the project with the idea of giving a restricted bi-color to the existed Catalan structures, and have included a selection of results in [18]. In the remainder of this section, we present a brief account of background for our investigation. The sequence of super-Catalan number was introcuced by Friedrich Wilhelm Karl Ernst Schröder in his paper [29] during his discusses on four “bracketing problems” and the term “Schröder number” seems to have been first used by Rogers [28]. Ernst Schöder gave the n-th super-Catalan number s(n) is the total number of bracketings of a string of n letters, but he did not mention any other combinatorial interpretations. While in 1994, David Hough discovered that the super-Catalan number were apparently known to Hipparchus in the second century B.C. (at least s(9) = 103049). The connection between bracketings and plane trees was known to Cayley [2]. The bijection between plane trees and polygon dissections appears in Etherington [15]. Currently, many good combinatorial structures enumerated by super-Catalan numbers are obtained by the references, from which we know the super-Catalan number not only counts the dissections of a convex polygon and plane trees, but also partitions (see [17]), various lattice paths (see [38]), permutations avoiding given patterns (see [8]), and so on. And in [36], Stanley narrated how the super-Catalan numbers are even more classical than has been believed before. He also recalled the three-term linear recurrence (see [6, 7, p.75])