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A co-ideal based identity-summand graph of a commutative semiring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2015)

Commentationes Mathematicae Universitatis Carolinae

Let I be a strong co-ideal of a commutative semiring R with identity. Let Γ I ( R ) be a graph with the set of vertices S I ( R ) = { x ∈ R ∖ I : x + y ∈ I for some y ∈ R ∖ I } , where two distinct vertices x and y are adjacent if and only if x + y ∈ I . We look at the diameter and girth of this graph. Also we discuss when Γ I ( R ) is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let p k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then p * k , m ( j , n ) = p k , m ( j , n ) .

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