304 research outputs found
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Sampling Hypergraphs with Given Degrees
There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm , and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant
Degree sequences of sufficiently dense random uniform hypergraphs
We find an asymptotic enumeration formula for the number of simple
-uniform hypergraphs with a given degree sequence, when the number of edges
is sufficiently large. The formula is given in terms of the solution of a
system of equations. We give sufficient conditions on the degree sequence which
guarantee existence of a solution to this system. Furthermore, we solve the
system and give an explicit asymptotic formula when the degree sequence is
close to regular. This allows us to establish several properties of the degree
sequence of a random -uniform hypergraph with a given number of edges. More
specifically, we compare the degree sequence of a random -uniform hypergraph
with a given number edges to certain models involving sequences of binomial or
hypergeometric random variables conditioned on their sum
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