Two topics in combinatorics: Generalized coinvariant algebras and Catalan-pair graphs
In this dissertation we study two combinatorial problems. The starting point of the first problem are coinvariant algebras, quotients of the polynomial ring in $n$ variables that serve as a remarkable connection between symmetric functions, representation theory and permutation statistics. Recently, inspired by the Delta Conjecture, generalized quotients were introduced, whose combinatorics are controlled by set partitions of a set of size $n$ into a given number of blocks. We exhibit quotients of the Stanley-Reisner ring of the Boolean algebra isomorphic to the given generalizations, extending an isomorphism known in the classical setting. Additionally, we introduce a quotient whose combinatorics are related to all set partitions of a given set, without any restrictions on the number of blocks.
Secondly, we look at Catalan numbers, a well-known combinatorial sequence with a variety of interpretations and applications. We study the interaction between two objects chosen from one of these interpretations, and represent this interaction in terms of a graph, also known as a bipartite circle graph. We introduce a random model to generate such graphs, and describe the asymptotic behaviour of various properties, including the number of edges, the number of isolated vertices, and its subgraphs.