# Cores with distinct parts and bigraded Fibonacci numbers

## TL;DR

This paper explores the enumeration and grading of specific core partitions related to Fibonacci and Catalan numbers, generalizing previous results and introducing new combinatorial structures.

## Contribution

It generalizes the enumeration of cores with distinct parts to (a,as+1)-cores and introduces a second grading on Fibonacci numbers via bigraded Catalan sequences.

## Key findings

- Number of (a,a+1)-cores with distinct parts equals Fibonacci number F_a.
- Generalization to (a,as+1)-cores with a natural grading.
- Introduction of a second grading on Fibonacci numbers using bigraded Catalan sequences.

## Abstract

The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with distinct parts are enumerated by another important sequence- Fibonacci numbers $F_a$. In this paper, we consider the abacus description of $(a,b)$-cores to introduce the natural grading and generalize this result to $(a,as+1)$-cores. We also use the bijection with Dyck paths to count the number of $(2k-1,2k+1)$-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence $C_{a,b} (q,t)$.