The Ungar Games on Graded Posets
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912568901107712 |
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| author | Paltrowitz, Jacob |
| author_facet | Paltrowitz, Jacob |
| contents | For a poset $P$, an Ungar move sends $P$ to $P\setminus T$, where $T$ is some subset of maximal elements of $P$. With these Ungar moves, Defant, Kravitz, and Williams define the Ungar games, where two players alternate making nontrivial Ungar moves until one player cannot make a move and loses. We characterize the second-player wins on graded posets. We first prove recursive characterizations of second-player wins before using these results to give classifications of the second-player wins in terms of boolean circuits. We also generalize Defant, Kravitz, and Williams' work on Young's Lattice $J(\mathbb{N}^2)$ to the higher-dimensional $J(\mathbb{N}^d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02959 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Ungar Games on Graded Posets Paltrowitz, Jacob Combinatorics 91A05, 91A46, 06B05, 06A07 For a poset $P$, an Ungar move sends $P$ to $P\setminus T$, where $T$ is some subset of maximal elements of $P$. With these Ungar moves, Defant, Kravitz, and Williams define the Ungar games, where two players alternate making nontrivial Ungar moves until one player cannot make a move and loses. We characterize the second-player wins on graded posets. We first prove recursive characterizations of second-player wins before using these results to give classifications of the second-player wins in terms of boolean circuits. We also generalize Defant, Kravitz, and Williams' work on Young's Lattice $J(\mathbb{N}^2)$ to the higher-dimensional $J(\mathbb{N}^d)$. |
| title | The Ungar Games on Graded Posets |
| topic | Combinatorics 91A05, 91A46, 06B05, 06A07 |
| url | https://arxiv.org/abs/2509.02959 |