A Factorization Theorem of Characteristic Polynomials
of Convex Geometries

M. Hachimori^{1}, M. Nakamura^{2}

nakamura@klee.c.u-tokyo.ac.jp

Annals of Combinatorics 11 (1) p.39-46 March, 2007

Abstract:

A convex geometry is a closure system whose closure operator satisfies the antiexchange
property. As is described in Sagan's survey paper, characteristic polynomials factorize
over nonnegative integers in several situations. We show that the characteristic polynomial of
a 2-tight convex geometry *K* factorizes over nonnegative integers if the clique complex of the
nbc-graph of *K* is pure and strongly connected. This factorization theorem is new in the sense
that it does not belong to any of the three categories mentioned in Sagan's survey.

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