Cyclic Sieving for Strong Dichotomy Enumeration
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910242539831296 |
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| author | Agustín-Aquino, Octavio A. |
| author_facet | Agustín-Aquino, Octavio A. |
| contents | Agustín-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as \emph{strong dichotomy classes}). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21658 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cyclic Sieving for Strong Dichotomy Enumeration Agustín-Aquino, Octavio A. Combinatorics 05A15, 05E18, 06A07, 20B25, 00A65 Agustín-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as \emph{strong dichotomy classes}). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general. |
| title | Cyclic Sieving for Strong Dichotomy Enumeration |
| topic | Combinatorics 05A15, 05E18, 06A07, 20B25, 00A65 |
| url | https://arxiv.org/abs/2605.21658 |