Enumeration of lattices of nullity k and containing r comparable reducible elements

2025/02/10 by A. N. Bhavale, Bhavale, A. N.
Computer Science · Mathematics · #06A05 #06A06 #06A07 #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2502.06912

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ≥ 4 elements, having nullity k ≥ 1, and containing 2 ≤ r ≤ 2k reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on n ≥ 4 elements, having nullity k ≥ 1. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on n elements.

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