3,240 research outputs found
Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles
Counting Arithmetical Structures on Paths and Cycles
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles
On connected sub-tree with fixed nodes in Cayley tree of Order 2
In this paper we found an exact formula for a finite sub-tree counting problem. Note that the formulas, which correspond to two extremal cases, are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general formula could be expressed as a linear combination of these Catalan triangles
Trickle-down processes and their boundaries
It is possible to represent each of a number of Markov chains as an evolving
sequence of connected subsets of a directed acyclic graph that grow in the
following way: initially, all vertices of the graph are unoccupied, particles
are fed in one-by-one at a distinguished source vertex, successive particles
proceed along directed edges according to an appropriate stochastic mechanism,
and each particle comes to rest once it encounters an unoccupied vertex.
Examples include the binary and digital search tree processes, the random
recursive tree process and generalizations of it arising from nested instances
of Pitman's two-parameter Chinese restaurant process, tree-growth models
associated with Mallows' phi model of random permutations and with
Schuetzenberger's non-commutative q-binomial theorem, and a construction due to
Luczak and Winkler that grows uniform random binary trees in a Markovian
manner. We introduce a framework that encompasses such Markov chains, and we
characterize their asymptotic behavior by analyzing in detail their Doob-Martin
compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment
Tree expansion in time-dependent perturbation theory
The computational complexity of time-dependent perturbation theory is
well-known to be largely combinatorial whatever the chosen expansion method and
family of parameters (combinatorial sequences, Goldstone and other Feynman-type
diagrams...). We show that a very efficient perturbative expansion, both for
theoretical and numerical purposes, can be obtained through an original
parametrization by trees and generalized iterated integrals. We emphasize above
all the simplicity and naturality of the new approach that links perturbation
theory with classical and recent results in enumerative and algebraic
combinatorics. These tools are applied to the adiabatic approximation and the
effective Hamiltonian. We prove perturbatively and non-perturbatively the
convergence of Morita's generalization of the Gell-Mann and Low wavefunction.
We show that summing all the terms associated to the same tree leads to an
utter simplification where the sum is simpler than any of its terms. Finally,
we recover the time-independent equation for the wave operator and we give an
explicit non-recursive expression for the term corresponding to an arbitrary
tree.Comment: 22 pages, 2 figure
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