On Graphical Partitions with Restricted Parts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908929270743040 |
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| author | Levy, Gilead |
| author_facet | Levy, Gilead |
| contents | An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed set. In this work, we investigate the probability that a uniformly random partition of an even integer $n$, subject to such restrictions, is graphical. We establish an upper bound on this probability expressed solely in terms of the Durfee square of the partition. Additionally, letting $p_g(n)$ denote the probability that a random restricted partition of an even integer $n$ is graphical, we prove that the limit inferior of $p_g(n)$ is 0. Furthermore, we obtain an explicit bound on the decay rate of $p_g(n)$ in terms of $n$ and the imposed restrictions on the parts. Our approach employs the Nash-Williams graphical condition, the saddle-point method and Edgeworth expansions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00007 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Graphical Partitions with Restricted Parts Levy, Gilead Number Theory Combinatorics 05A17, 05C07, 11P81, 11P82 An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed set. In this work, we investigate the probability that a uniformly random partition of an even integer $n$, subject to such restrictions, is graphical. We establish an upper bound on this probability expressed solely in terms of the Durfee square of the partition. Additionally, letting $p_g(n)$ denote the probability that a random restricted partition of an even integer $n$ is graphical, we prove that the limit inferior of $p_g(n)$ is 0. Furthermore, we obtain an explicit bound on the decay rate of $p_g(n)$ in terms of $n$ and the imposed restrictions on the parts. Our approach employs the Nash-Williams graphical condition, the saddle-point method and Edgeworth expansions. |
| title | On Graphical Partitions with Restricted Parts |
| topic | Number Theory Combinatorics 05A17, 05C07, 11P81, 11P82 |
| url | https://arxiv.org/abs/2510.00007 |