3,031 research outputs found
On the effective and automatic enumeration of polynomial permutation classes
We describe an algorithm, implemented in Python, which can enumerate any
permutation class with polynomial enumeration from a structural description of
the class. In particular, this allows us to find formulas for the number of
permutations of length n which can be obtained by a finite number of block
sorting operations (e.g., reversals, block transpositions, cut-and-paste
moves)
The effect of negative feedback loops on the dynamics of Boolean networks
Feedback loops in a dynamic network play an important role in determining the
dynamics of that network. Through a computational study, in this paper we show
that networks with fewer independent negative feedback loops tend to exhibit
more regular behavior than those with more negative loops. To be precise, we
study the relationship between the number of independent feedback loops and the
number and length of the limit cycles in the phase space of dynamic Boolean
networks. We show that, as the number of independent negative feedback loops
increases, the number (length) of limit cycles tends to decrease (increase).
These conclusions are consistent with the fact, for certain natural biological
networks, that they on the one hand exhibit generally regular behavior and on
the other hand show less negative feedback loops than randomized networks with
the same numbers of nodes and connectivity
Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements
Monotone path polytopes arise as a special case of the construction of fiber
polytopes, introduced by Billera and Sturmfels. A simple example is provided by
the permutahedron, which is a monotone path polytope of the standard unit cube.
The permutahedron is the zonotope polar to the braid arrangement. We show how
the zonotopes polar to the cones of certain deformations of the braid
arrangement can be realized as monotone path polytopes. The construction is an
extension of that of the permutahedron and yields interesting connections
between enumerative combinatorics of hyperplane arrangements and geometry of
monotone path polytopes
Monotone Hurwitz numbers in genus zero
Hurwitz numbers count branched covers of the Riemann sphere with specified
ramification data, or equivalently, transitive permutation factorizations in
the symmetric group with specified cycle types. Monotone Hurwitz numbers count
a restricted subset of the branched covers counted by the Hurwitz numbers, and
have arisen in recent work on the the asymptotic expansion of the
Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study
of monotone Hurwitz numbers. We prove two results that are reminiscent of those
for classical Hurwitz numbers. The first is the monotone join-cut equation, a
partial differential equation with initial conditions that characterizes the
generating function for monotone Hurwitz numbers in arbitrary genus. The second
is our main result, in which we give an explicit formula for monotone Hurwitz
numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic
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