Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915145371877376 |
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| author | Bhavale, A. N. |
| author_facet | Bhavale, A. N. |
| contents | In 2002 Thakare et al.\ counted non-isomorphic lattices on $n$ elements, having nullity up to two. In 2020 Bhavale and Waphare introduced the concept of RC-lattices as the class of all lattices in which all the reducible elements are comparable. In this paper, we enumerate all non-isomorphic RC-lattices on $n$ elements. For this purpose, firstly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$, and containing $2 \leq r \leq 2k$ reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on $n$ elements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements Bhavale, A. N. Combinatorics 06A05, 06A06, 06A07 In 2002 Thakare et al.\ counted non-isomorphic lattices on $n$ elements, having nullity up to two. In 2020 Bhavale and Waphare introduced the concept of RC-lattices as the class of all lattices in which all the reducible elements are comparable. In this paper, we enumerate all non-isomorphic RC-lattices on $n$ elements. For this purpose, firstly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$, and containing $2 \leq r \leq 2k$ reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on $n$ elements. |
| title | Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements |
| topic | Combinatorics 06A05, 06A06, 06A07 |
| url | https://arxiv.org/abs/2502.06912 |