44 research outputs found
Abelian networks IV. Dynamics of nonhalting networks
An abelian network is a collection of communicating automata whose state
transitions and message passing each satisfy a local commutativity condition.
This paper is a continuation of the abelian networks series of Bond and Levine
(2016), for which we extend the theory of abelian networks that halt on all
inputs to networks that can run forever. A nonhalting abelian network can be
realized as a discrete dynamical system in many different ways, depending on
the update order. We show that certain features of the dynamics, such as
minimal period length, have intrinsic definitions that do not require
specifying an update order.
We give an intrinsic definition of the \emph{torsion group} of a finite
irreducible (halting or nonhalting) abelian network, and show that it coincides
with the critical group of Bond and Levine (2016) if the network is halting. We
show that the torsion group acts freely on the set of invertible recurrent
components of the trajectory digraph, and identify when this action is
transitive.
This perspective leads to new results even in the classical case of sinkless
rotor networks (deterministic analogues of random walks). In Holroyd et. al
(2008) it was shown that the recurrent configurations of a sinkless rotor
network with just one chip are precisely the unicycles (spanning subgraphs with
a unique oriented cycle, with the chip on the cycle). We generalize this result
to abelian mobile agent networks with any number of chips. We give formulas for
generating series such as where is the number of recurrent chip-and-rotor configurations with
chips; is the diagonal matrix of outdegrees, and is the adjacency
matrix. A consequence is that the sequence completely
determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure
Computational complexity of counting coincidences
Can you decide if there is a coincidence in the numbers counting two
different combinatorial objects? For example, can you decide if two regions in
have the same number of domino tilings? There are two versions
of the problem, with and boxes. We
prove that in both cases the coincidence problem is not in the polynomial
hierarchy unless the polynomial hierarchy collapses to a finite level. While
the conclusions are the same, the proofs are notably different and generalize
in different directions.
We proceed to explore the coincidence problem for counting independent sets
and matchings in graphs, matroid bases, order ideals and linear extensions in
posets, permutation patterns, and the Kronecker coefficients. We also make a
number of conjectures for counting other combinatorial objects such as plane
triangulations, contingency tables, standard Young tableaux, reduced
factorizations and the Littlewood--Richardson coefficients.Comment: 23 pages, 6 figure
Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields
A maximal minor of the Laplacian of an -vertex Eulerian digraph
gives rise to a finite group
known as the sandpile (or critical) group of . We determine
of the generalized de Bruijn graphs with
vertices and arcs for and , and closely related generalized Kautz graphs, extending and
completing earlier results for the classical de Bruijn and Kautz graphs.
Moreover, for a prime and an -cycle permutation matrix
we show that is isomorphic to the
quotient by of the centralizer of in
. This offers an explanation for the coincidence of
numerical data in sequences A027362 and A003473 of the OEIS, and allows one to
speculate upon a possibility to construct normal bases in the finite field
from spanning trees in .Comment: I+24 page
Extensions of the Kahn--Saks inequality for posets of width two
The Kahn--Saks inequality is a classical result on the number of linear
extensions of finite posets. We give a new proof of this inequality for posets
of width two using explicit injections of lattice paths. As a consequence we
obtain a -analogue, a multivariate generalization and an equality condition
in this case. We also discuss the equality conditions of the Kahn--Saks
inequality for general posets and prove several implications between conditions
conjectured to be equivalent.Comment: 25 pages; v3: added conditions to Theorem 1.4 and 1.6, revised
Conjecture in Section 8, added example 1.