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Some matroid inequalities

✍ Scribed by Anders Björner

Publisher
Elsevier Science
Year
1980
Tongue
English
Weight
202 KB
Volume
31
Category
Article
ISSN
0012-365X

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✦ Synopsis

For a given matroid M of rank r we will be interested in its Whitney numbers (of the first kind) wo, wl, l -9 w, and its independence numbers IO, J1, . . . , I,. Here wk is the h'-k-coefficient in the characteristic polynomial *&$(A) =I i (-I)'w,h'-' i=O of M and Ik is the number of independent k-element sets in M. Note that W, = F(M), wher c /i(M) is the absolute value of the Miibius function p computed on the lattice of flats of M, and I, = b(M), the number of bases or complexity of M. Let C~ = min{(C( 1 C is a circuit in M). A..-

We say that M is loopless if c, > 1 and that M is a (combinatorial) geometry if c, > 2. In the sequel the letters r and n will always refer to the rank and cardinality respectively of the matroid being considered.

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