{"title":"Arithmetic and Asymptotic Properties for Some Functions Related to the Least r-Gaps of Partitions","authors":"Nayandeep Deka Baruah, Pranjal Talukdar","doi":"10.1007/s00026-025-00772-z","DOIUrl":null,"url":null,"abstract":"<div><p>The minimal excludant or mex of a partition, as introduced by Andrews and Newman [3], is the smallest positive integer missing from that partition. Recently, Ballantine and Merca [5] studied a generalization of mex called the least <i>r</i>-gap and defined as the smallest positive integer which does not appear at least <i>r</i> times in the partition. In this article, we deduce the generating functions for certain arithmetic functions related to the least <i>r</i>-gaps and establish their connections with known partition functions. We also obtain arithmetic properties and asymptotic formulae for some of these functions. As a consequence of one such result, we find an asymptotic formula for the Andrews’ singular overpartition function <span>\\({\\overline{C}}_{k,i}(n)\\)</span>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"30 2","pages":"525 - 551"},"PeriodicalIF":0.8000,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-025-00772-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The minimal excludant or mex of a partition, as introduced by Andrews and Newman [3], is the smallest positive integer missing from that partition. Recently, Ballantine and Merca [5] studied a generalization of mex called the least r-gap and defined as the smallest positive integer which does not appear at least r times in the partition. In this article, we deduce the generating functions for certain arithmetic functions related to the least r-gaps and establish their connections with known partition functions. We also obtain arithmetic properties and asymptotic formulae for some of these functions. As a consequence of one such result, we find an asymptotic formula for the Andrews’ singular overpartition function \({\overline{C}}_{k,i}(n)\).
期刊介绍：
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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