2011/09/06 by Peter J. Cameron, Cameron, Peter J., Maximilien Gadouleau +3
Mathematics · #05B15 #05B35 (Primary) #94A17 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B15 #msc:05B35 #msc:94A17
paper · pdf · doi:10.48550/arxiv.1109.1216
23pp. Submitted to J. Combinatorial Theory (Series A)
arxiv created 2011/09/06 · arxiv updated 2011/09/07
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a rank operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows.