Counting of lattices containing up to four comparable reducible elements and having nullity up to three
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| Format: | Preprint |
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2024
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| _version_ | 1866917957684166656 |
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| author | Aware, B. P. Bhavale, A. N. |
| author_facet | Aware, B. P. Bhavale, A. N. |
| contents | In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. According to Bhavale and Waphare, if a dismantlable lattice of nullity k contains r reducible elements then 2 $\leq$ r $\leq$ 2k. In 2003 Pawar and Waphare counted all non-isomorphic lattices with equal number of elements and edges, which are precisely the lattices of nullity one. Recently, Bhavale and Aware counted all non-isomorphic lattices on n elements having nullity up to two. Bhavale and Aware also counted all non-isomorphic lattices on n elements, containing up to three reducible elements, having nullity k $\geq$ 2. In this paper, we count up to isomorphism the class of all lattices on n elements containing four comparable reducible elements, and having nullity three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03627 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting of lattices containing up to four comparable reducible elements and having nullity up to three Aware, B. P. Bhavale, A. N. Combinatorics 06A07, 06A06 In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. According to Bhavale and Waphare, if a dismantlable lattice of nullity k contains r reducible elements then 2 $\leq$ r $\leq$ 2k. In 2003 Pawar and Waphare counted all non-isomorphic lattices with equal number of elements and edges, which are precisely the lattices of nullity one. Recently, Bhavale and Aware counted all non-isomorphic lattices on n elements having nullity up to two. Bhavale and Aware also counted all non-isomorphic lattices on n elements, containing up to three reducible elements, having nullity k $\geq$ 2. In this paper, we count up to isomorphism the class of all lattices on n elements containing four comparable reducible elements, and having nullity three. |
| title | Counting of lattices containing up to four comparable reducible elements and having nullity up to three |
| topic | Combinatorics 06A07, 06A06 |
| url | https://arxiv.org/abs/2412.03627 |