317 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.
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
Enumerative and asymptotic analysis of a moduli space
We focus on combinatorial aspects of the Hilbert series of the cohomology
ring of the moduli space of stable pointed curves of genus zero. We show its
graded Hilbert series satisfies an integral operator identity. This is used to
give asymptotic behavior, and in some cases, exact values, of the coefficients
themselves. We then study the total dimension, that is, the sum of the
coefficients of the Hilbert series. Its asymptotic behavior involves the
Lambert W function, which has applications to classical tree enumeration,
signal processing and fluid mechanics.Comment: 14 page
Linear algebraic techniques for spanning tree enumeration
Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in
a finite graph can be computed from the determinant of any of its reduced
Laplacian matrices. In many cases, even for well-studied families of graphs,
this can be computationally or algebraically taxing. We show how two well-known
results from linear algebra, the Matrix Determinant Lemma and the Schur
complement, can be used to elegantly count the spanning trees in several
significant families of graphs.Comment: This paper presents unweighted versions of the results in
arXiv:1903.03575 with more concrete and concise proofs. It is intended for a
broad audience and has extra emphasis on exposition. It will appear in the
American Mathematical Monthl
A Family of matroid intersection algorithms for the computation of approximated symbolic network functions
In recent years, the technique of simplification during generation has turned out to be very promising for the efficient computation of approximate symbolic network functions for large transistor circuits. In this paper it is shown how symbolic network functions can be simplified during their generation with any well-known symbolic network analysis method. The underlying algorithm for the different techniques is always a matroid intersection algorithm. It is shown that the most efficient technique is the two-graph method. An implementation of the simplification during generation technique with the two-graph method illustrates its benefits for the symbolic analysis of large analog circuits
Combinatorial families of multilabelled increasing trees and hook-length formulas
In this work we introduce and study various generalizations of the notion of
increasingly labelled trees, where the label of a child node is always larger
than the label of its parent node, to multilabelled tree families, where the
nodes in the tree can get multiple labels. For all tree classes we show
characterizations of suitable generating functions for the tree enumeration
sequence via differential equations. Furthermore, for several combinatorial
classes of multilabelled increasing tree families we present explicit
enumeration results. We also present multilabelled increasing tree families of
an elliptic nature, where the exponential generating function can be expressed
in terms of the Weierstrass-p function or the lemniscate sine function.
Furthermore, we show how to translate enumeration formulas for multilabelled
increasing trees into hook-length formulas for trees and present a general
"reverse engineering" method to discover hook-length formulas associated to
such tree families.Comment: 37 page
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