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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.00903 |
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| _version_ | 1866912844707004416 |
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| author | Kutin, Samuel A. Smithline, Lawren M. |
| author_facet | Kutin, Samuel A. Smithline, Lawren M. |
| contents | We introduce a guessing game, permutation Wordle, in which a guesser attempts to recover a hidden permutation in $S_n$. In each round, the guesser guesses a permutation (using information from previous rounds) and is told which entries of that permutation are correct. We describe a natural guessing strategy, which we believe to be optimal. We show that the number of permutations this strategy solves in $k+1$ rounds is the Eulerian number $A(n,k)$.
We also describe an extension to suited permutations: the setter chooses a permutation in $S_n$ and also a coloring of $[n]$ using $s$ colors. We generalize our strategy, give a recurrence for the number of suited permutations solved in $k+1$ rounds, and relate these numbers to the Eulerian numbers. In the case of two suits, or signed permutations, we also relate these numbers to the Eulerian numbers of type B. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Permutation Wordle Kutin, Samuel A. Smithline, Lawren M. Combinatorics 05A05 We introduce a guessing game, permutation Wordle, in which a guesser attempts to recover a hidden permutation in $S_n$. In each round, the guesser guesses a permutation (using information from previous rounds) and is told which entries of that permutation are correct. We describe a natural guessing strategy, which we believe to be optimal. We show that the number of permutations this strategy solves in $k+1$ rounds is the Eulerian number $A(n,k)$. We also describe an extension to suited permutations: the setter chooses a permutation in $S_n$ and also a coloring of $[n]$ using $s$ colors. We generalize our strategy, give a recurrence for the number of suited permutations solved in $k+1$ rounds, and relate these numbers to the Eulerian numbers. In the case of two suits, or signed permutations, we also relate these numbers to the Eulerian numbers of type B. |
| title | Permutation Wordle |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2408.00903 |