9,557 research outputs found
A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
We present a version of the weighted cellular matrix-tree theorem that is
suitable for calculating explicit generating functions for spanning trees of
highly structured families of simplicial and cell complexes. We apply the
result to give weighted generalizations of the tree enumeration formulas of
Adin for complete colorful complexes, and of Duval, Klivans and Martin for
skeleta of hypercubes. We investigate the latter further via a logarithmic
generating function for weighted tree enumeration, and derive another
tree-counting formula using the unsigned Euler characteristics of skeleta of a
hypercube and the Crapo -invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified
and condensed. Final version to appear in J. Combin. Theory Ser.
On the Hausdorff dimension of the Rauzy gasket
In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is
less than 2. By this result, we answer a question addressed by Pierre Arnoux.
Also, this question is a very particular case of the conjecture stated by S.P.
Novikov and A. Ya. Maltsev in 2003.Comment: 23 pages, 5 figure
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