4,784 research outputs found
Permutations destroying arithmetic progressions in finite cyclic groups
A permutation \pi of an abelian group G is said to destroy arithmetic
progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that
is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an
AP. In a paper from 2004, the first author conjectured that such a permutation
exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case
of a more general result, that such a permutation exists for all n >= n_0, for
some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct
such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).Comment: 11 pages, no figure
Crucial and bicrucial permutations with respect to arithmetic monotone patterns
A pattern is a permutation, and an arithmetic occurrence of in
(another) permutation is a subsequence
of that is order isomorphic to
where the numbers form an arithmetic progression. A
permutation is -crucial if it avoids arithmetically the patterns
and but its extension to the right by any element
does not avoid arithmetically these patterns. A -crucial permutation
that cannot be extended to the left without creating an arithmetic occurrence
of or is called -bicrucial.
In this paper we prove that arbitrary long -crucial and
-bicrucial permutations exist for any . Moreover, we
show that the minimal length of a -crucial permutation is
, while the minimal length of a
-bicrucial permutation is at most ,
again for
Number of cycles in the graph of 312-avoiding permutations
The graph of overlapping permutations is defined in a way analogous to the De
Bruijn graph on strings of symbols. That is, for every permutation there is a directed edge from the
standardization of to the standardization of
. We give a formula for the number of cycles of
length in the subgraph of overlapping 312-avoiding permutations. Using this
we also give a refinement of the enumeration of 312-avoiding affine
permutations and point out some open problems on this graph, which so far has
been little studied.Comment: To appear in the Journal of Combinatorial Theory - Series
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
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