633 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
Extremal problems for ordered hypergraphs: small patterns and some enumeration
We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum
numbers of edges and maximum numbers of vertex-edge incidences in simple
hypergraphs H which have n vertices and do not contain a fixed hypergraph F;
the containment respects linear orderings of vertices. We determine both
functions exactly if F has only distinct singleton edges or if F is one of the
55 hypergraphs with at most four incidences (we give proofs only for six
cases). We prove some exact formulae and recurrences for the numbers of
hypergraphs, simple and all, with n incidences and derive rough logarithmic
asymptotics of these numbers. Identities analogous to Dobinski's formula for
Bell numbers are given.Comment: 22 pages, submitted to Discrete Applied Mathematic
Hereditary properties of partitions, ordered graphs and ordered hypergraphs
In this paper we use the Klazar-Marcus-Tardos method to prove that if a
hereditary property of partitions P has super-exponential speed, then for every
k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k},
where 1 <= i <= k. We also prove a similar jump, from exponential to factorial,
in the possible speeds of monotone properties of ordered graphs, and of
hereditary properties of ordered graphs not containing large complete, or
complete bipartite ordered graphs.
Our results generalize the Stanley-Wilf Conjecture on the number of
n-permutations avoiding a fixed permutation, which was recently proved by the
combined results of Klazar and of Marcus and Tardos. Our main results follow
from a generalization to ordered hypergraphs of the theorem of Marcus and
Tardos.Comment: 25 pgs, no figure
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