On the number of missing integers in partitions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914471220346880 |
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| author | Bhoria, Subhash Chand Eyyunni, Pramod Santra, Subhrangsu |
| author_facet | Bhoria, Subhash Chand Eyyunni, Pramod Santra, Subhrangsu |
| contents | In the preceding decade, Andrews and Newman resurrected the concept of a `minimal excludant' of a partition ($mex$, for short), namely, the least positive missing integer in a partition. Subsequently, several authors have not only studied its generalizations, analogues and the like but also connected the mex to several important partition statistics. In the present paper, we study the set of missing positive integers as a whole, in two different classes of partitions, namely, unrestricted partitions and overpartitions. To be precise, a $missing \ integer$ is a positive integer that is less than the largest part of a partition and which does not occur as a part. In particular, we examine the number of partitions with a given number of missing integers, determine congruences for two pairs of functions associated to them, and propose three bias type inequality conjectures for these functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12557 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the number of missing integers in partitions Bhoria, Subhash Chand Eyyunni, Pramod Santra, Subhrangsu Combinatorics Number Theory 11P81, 11P83, 11P84, 05A17 In the preceding decade, Andrews and Newman resurrected the concept of a `minimal excludant' of a partition ($mex$, for short), namely, the least positive missing integer in a partition. Subsequently, several authors have not only studied its generalizations, analogues and the like but also connected the mex to several important partition statistics. In the present paper, we study the set of missing positive integers as a whole, in two different classes of partitions, namely, unrestricted partitions and overpartitions. To be precise, a $missing \ integer$ is a positive integer that is less than the largest part of a partition and which does not occur as a part. In particular, we examine the number of partitions with a given number of missing integers, determine congruences for two pairs of functions associated to them, and propose three bias type inequality conjectures for these functions. |
| title | On the number of missing integers in partitions |
| topic | Combinatorics Number Theory 11P81, 11P83, 11P84, 05A17 |
| url | https://arxiv.org/abs/2604.12557 |