Asymptotics for moments of the minimal partition excludant in congruence classes

TL;DR

This paper derives asymptotic formulas for the moments of generalized minimal excludants in specific congruence classes, revealing their asymptotic equivalence for fixed moduli, advancing understanding of partition statistics.

Contribution

It introduces asymptotic formulas for moments of minimal excludants in congruence classes, extending previous work on the minimal excludant statistic.

Findings

Moments of minimal excludants in fixed moduli are asymptotically equal.

Established asymptotic formulas for generalized minimal excludants.

Revealed asymptotic behavior of partition-related statistics.

Abstract

The minimal excludant statistic, which denotes the smallest positive integer that is not a part of an integer partition, has received great interest in recent years. In this paper, we move on to the smallest positive integer whose frequency is less than a given number. We establish an asymptotic formula for the moments of such generalized minimal excludants that fall in a specific congruence class. In particular, our estimation reveals that the moments associated with a fixed modulus are asymptotically ``equal''.