A permutation pattern is a sub-permutation of a longer permutation. The study of permutation pattern came from enumerative combinations and focused on finding formulas for the number of permutation which avoids a fixed and typically short permutation. The permutation π, written as a word in one line notation is said to contain the permutation σ, if there exists a subsequence of entries of σ that as the same relative order as π , and in this case σ is said to be a pattern of π , written as σ . Otherwise π is said to avoid the permutation σ. This paper investigated some algebraic theoretical properties of Aunu permutations pattern of the (123) - avoiding class in relation to lattices and graph structure. The properties obtained were identified using the direct product of two sets of Aunu numbers of 5-sample elements. The paper further established some graph structural models of different Eulerian circuits using the adjacency matrix of the (123) - avoiding class of these special numbers.
Key words: Aunu numbers, Permutation avoiding, partial order, natural ordering, direct product, lattices and graphs.
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