Abstract
Given a family A of subsets of an ordered set E, we define a construction giving a family of sets B in 1–1 correspondence with A; the same construction applied to B then gives A. For each subset X of E, \(A \cap B \subseteq X \subseteq A \cup B\) for exactly one pair of A∈A and corresponding B∈B. When the family B is the basis collection of a matroid on E, A can be described simply in terms of the matroid structure. A polynomial is defined which, in this latter case, is the Tutte polynomial of the matroid.
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References
H.H. Crapo, The Tutte polynomial, Aeq. Math. 3 (1969), 211–229.
J.E. Dawson, Optimal matroid bases: an algorithm based on cocircuits, Quart. J. Math. (Oxford) (2) 31 (1980), 65–69.
J.E. Dawson, A simple approach to some basic results in matroid theory. J. Math. Anal. Appl., 75 (1980), 611–615.
R. von Randow, Introduction to the Theory of Matroids, Lecture Notes in Economics and Mathematical Systems 109, Springer-Verlag, 1975.
D.J.A. Welsh, Kruskal's theorem for matroids, Proc. Camb. Phil. Soc. 64 (1968), 3–4.
D.J.A. Welsh, Matroid Theory, Academic Press, New York, 1976.
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© 1981 Springer-Verlag
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Dawson, J.E. (1981). A construction for a family of sets and its application to matroids. In: McAvaney, K.L. (eds) Combinatorial Mathematics VIII. Lecture Notes in Mathematics, vol 884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091815
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DOI: https://doi.org/10.1007/BFb0091815
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