{ "id": "1905.06304", "version": "v1", "published": "2019-05-15T17:25:52.000Z", "updated": "2019-05-15T17:25:52.000Z", "title": "Partitions and the maximal excludant", "authors": [ "Shane Chern" ], "categories": [ "math.CO" ], "abstract": "For each nonempty integer partition $\\pi$, we define the maximal excludant of $\\pi$ to be the largest nonnegative integer smaller than the largest part of $\\pi$ that is not a part of $\\pi$. Let $\\sigma\\!\\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\\sigma\\!\\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews \\textit{et al.} and Cohen. Further, we show that, as $n\\to \\infty$, $\\sigma\\!\\operatorname{maex}(n)$ is asymptotic to the sum of largest parts of all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\\to \\infty$.", "revisions": [ { "version": "v1", "updated": "2019-05-15T17:25:52.000Z" } ], "analyses": { "subjects": [ "05A17", "05A19", "11P84" ], "keywords": [ "maximal excludant", "largest part", "nonempty integer partition", "largest nonnegative integer smaller", "mock theta function" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }