Universal Matrices for Counting Fibonomial and $C$-nomial Coefficients by their $p$-adic Valuations

TL;DR

This paper generalizes matrix product formulas for counting binomial and $C$-nomial coefficients by their $p$-adic valuations, revealing universal matrices applicable to a broad class of sequences including Fibonomial coefficients.

Contribution

It introduces universal matrix formulas for $C$-nomial coefficients with strong divisibility sequences, extending Rowland's work and applying to Fibonomial coefficients.

Findings

Universal matrices are independent of the specific sequence $C$.

The formulas extend to $C$-multinomial coefficients.

The approach unifies counting methods for various combinatorial coefficients.

Abstract

Rowland found a matrix product formula for generating functions counting binomial coefficients by their $p$-adic valuations. A natural generalization of binomial coefficients was introduced by Knuth and Wilf defined by a sequence $C$. We obtain analogous matrix product formulas counting these $C$-nomial coefficients when $C$ is a strong divisibility sequence. Surprisingly, the matrices are universal in the sense that they are independent of $C$. We further extend this product to $C$-multinomial coefficients.