59 research outputs found
Limits of permutation sequences
A permutation sequence is said to be convergent if the density of occurrences
of every fixed permutation in the elements of the sequence converges. We prove
that such a convergent sequence has a natural limit object, namely a Lebesgue
measurable function with the additional properties that,
for every fixed , the restriction is a cumulative
distribution function and, for every , the restriction
satisfies a "mass" condition. This limit process is well-behaved:
every function in the class of limit objects is a limit of some permutation
sequence, and two of these functions are limits of the same sequence if and
only if they are equal almost everywhere. An ingredient in the proofs is a new
model of random permutations, which generalizes previous models and might be
interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory,
Series B. arXiv admin note: text overlap with arXiv:1106.166
Quasirandom permutations are characterized by 4-point densities
For permutations Ď and Ď of lengths |Ď|â¤|Ď| , let t(Ď,Ď) be the probability that the restriction of Ď to a random |Ď| -point set is (order) isomorphic to Ď . We show that every sequence {Ďj} of permutations such that |Ďj|ââ and t(Ď,Ďj)â1/4! for every 4-point permutation Ď is quasirandom (that is, t(Ď,Ďj)â1/|Ď|! for every Ď ). This answers a question posed by Graham
Limits of Structures and the Example of Tree-Semilattices
The notion of left convergent sequences of graphs introduced by Lov\' asz et
al. (in relation with homomorphism densities for fixed patterns and
Szemer\'edi's regularity lemma) got increasingly studied over the past
years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general
framework for convergence of sequences of structures. In particular, the
authors introduced the notion of -convergence, which is a natural
generalization of left-convergence. In this paper, we initiate study of
-convergence for structures with functional symbols by focusing on the
particular case of tree semi-lattices. We fully characterize the limit objects
and give an application to the study of left convergence of -partite
cographs, a generalization of cographs
Finitely forcible graphons and permutons
We investigate when limits of graphs (graphons) and permutations (permutons)
are uniquely determined by finitely many densities of their substructures,
i.e., when they are finitely forcible. Every permuton can be associated with a
graphon through the notion of permutation graphs. We find permutons that are
finitely forcible but the associated graphons are not. We also show that all
permutons that can be expressed as a finite combination of monotone permutons
and quasirandom permutons are finitely forcible, which is the permuton
counterpart of the result of Lovasz and Sos for graphons.Comment: 30 pages, 18 figure
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