Enumerative combinatorics package for CoCoA

TL;DR

This paper describes a novel CoCoA package named combinatorics that computes a broad class of enumerative invariants, including permutations, Catalan generalizations, and related number sequences. It leverages existing CoCoA primitives ($e.g.$, binomial coefficients, permutations) and extends the library with modules for pattern-avoiding permutations, super Catalan numbers, Fuss-Catalan numbers, Schröder numbers, Narayana numbers, Stirling numbers, Bell numbers, and factorial/primorial primes. The authors present concrete API examples and outputs to illustrate function names such as AvoidingPermutations, CatalanNumber, SuperCatalanNumber, NarayanaNumber, FactorialPrime, PrimorialPrime, RisingFactorial, BellNumber, and PrintPascalTriangle, demonstrating seamless integration with CoCoA’s $QQ[x]$ environment. This work enhances CoCoA’s utility for researchers in enumerative combinatorics by providing a cohesive, extensible toolkit for computing a wide range of invariants and their interrelationships.

Abstract

We introduce the package combinatorics for the software CoCoA. This package provides a data structure and the necessary methods for computing several known enumerative combinatorial invariants.