On Some Generalisations of Gauss Sequences

TL;DR

This paper introduces Euler-Gauss sequences, generalizes Gauss sequences, explores their properties, and establishes new cyclic sieving phenomena, expanding understanding of sequence congruences and combinatorial interpretations.

Contribution

It defines Euler-Gauss sequences, compares them with Gauss sequences, and introduces q-analogs with new CSP conditions, broadening sequence theory and combinatorial applications.

Findings

Euler-Gauss sequences include sequences based on prime factors.

q-Euler-Gauss sequences satisfy standard CSP with restrictions.

New CSP conditions for SPF and GPF sequences are established.

Abstract

In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. Unlike the Gauss sequences, this extended class also contains sequences based on distinct prime factors. In particular, the sequences Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at 1), extensively studied by prominent mathematicians such as Erdos and Alladi, arise as examples of this class of Euler-Gauss sequences and not in the class of Gauss sequences. In the latter part of the paper, we obtain q-analogs of the defined Euler-Gauss sequences and establish characteristic properties that reveal their structure and fill gaps in the literature on q-Gauss sequences. In recent works, q-Gauss sequences have been shown to admit interesting combinatorial interpretations and to exhibit the Cyclic Sieving Phenomenon (CSP). Not only do our q-Euler-Gauss sequences satisfy the standard CSP with some restriction, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.