A New Approach to Enumerating Statistics Modulo $n$

TL;DR

This paper introduces a novel method for modular enumeration of polynomials modulo $x^n-1$, simplifying complex problems by leveraging residues modulo cyclotomic polynomials, and applies it to solve open problems in $q$-multinomial coefficients and $q$-Catalan numbers.

Contribution

The paper presents a new approach to compute polynomial remainders modulo $x^n-1$ using residues modulo $ ext{Φ}_d(x)$, solving several open problems in modular enumeration.

Findings

Simplified computation of polynomial remainders using residues modulo cyclotomic polynomials.

Solved an open problem on remainders of $q$-multinomial coefficients.

Discovered a cyclic group operation on lattice paths related to $q$-Catalan numbers.

Abstract

We find a new approach to computing the remainder of a polynomial modulo $x^n-1$; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative $\mathbb{Q}$-algebra, our first main result constructs the remainder simply from the coefficients of residues of the polynomial modulo $\Phi_d(x)$ for each $d\mid n$. Since such residues can often be found to have nice values, this simplifies a number of modular enumeration problems; indeed in some cases, such residues are already known while the related modular enumeration problem has remained unsolved. We list six such cases which our technique makes easy to solve. Our second main result is a formula for the unique polynomial $a$ such that $a \equiv f \mod \Phi_n(x)$ and $a\equiv 0 \mod x^d-1$ for each proper divisor $d$ of $n$. We find a formula for remainders of $q$-multinomial coefficients and for remainders of $q$-Catalan numbers modulo $q^n-1$, reducing each problem to a finite number of cases for any fixed $n$. In the prior case, we solve an open problem posed by Hartke and Radcliffe. In considering $q$-Catalan numbers modulo $q^n-1$, we discover a cyclic group operation on certain lattice paths which behaves predictably with regard to major index. We also make progress on a problem in modular enumeration on subset sums posed by Kitchloo and Pachter.