122 research outputs found
Counting packings of generic subsets in finite groups
A packing of subsets in a group is a
sequence such that are
disjoint subsets of . We give a formula for the number of packings if the
group is finite and if the subsets satisfy
a genericity condition. This formula can be seen as a generalization of the
falling factorials which encode the number of packings in the case where all
the sets are singletons
Universal and Near-Universal Cycles of Set Partitions
We study universal cycles of the set of -partitions of the
set and prove that the transition digraph associated
with is Eulerian. But this does not imply that universal cycles
(or ucycles) exist, since vertices represent equivalence classes of partitions!
We use this result to prove, however, that ucycles of exist for
all when . We reprove that they exist for odd when and that they do not exist for even when . An infinite family
of for which ucycles do not exist is shown to be those pairs for which
is odd (). We also show that there exist
universal cycles of partitions of into subsets of distinct sizes when
is sufficiently smaller than , and therefore that there exist universal
packings of the partitions in . An analogous result for
coverings completes the investigation.Comment: 22 page
On Universal Cycles for new Classes of Combinatorial Structures
A universal cycle (u-cycle) is a compact listing of a collection of
combinatorial objects. In this paper, we use natural encodings of these objects
to show the existence of u-cycles for collections of subsets, matroids,
restricted multisets, chains of subsets, multichains, and lattice paths. For
subsets, we show that a u-cycle exists for the -subsets of an -set if we
let vary in a non zero length interval. We use this result to construct a
"covering" of length for all subsets of of size
exactly with a specific formula for the term. We also show that
u-cycles exist for all -length words over some alphabet which
contain all characters from Using this result we provide
u-cycles for encodings of Sperner families of size 2 and proper chains of
subsets
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
Fat 4-polytopes and fatter 3-spheres
We introduce the fatness parameter of a 4-dimensional polytope P, defined as
\phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in
4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes
bounded?
We describe and analyze a hyperbolic geometry construction that produces
4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of
2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite
covering spaces of surfaces, we show that fatness is not bounded for the more
general class of strongly regular CW decompositions of the 3-sphere.Comment: 12 pages, 12 figures. This version has minor changes proposed by the
second refere
The existence of k-radius sequences
Let and be positive integers, and let be an alphabet of size .
A sequence over of length is a \emph{-radius sequence} if any two
distinct elements of occur within distance of each other somewhere in
the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in
order to produce an efficient caching strategy when computing certain functions
on large data sets such as medical images.
Let be the length of the shortest -ary -radius sequence. The
paper shows, using a probabilistic argument, that whenever is fixed and
The paper observes that the same argument generalises to the situation when
we require the following stronger property for some integer such that
: any distinct elements of must simultaneously occur
within a distance of each other somewhere in the sequence.Comment: 8 pages. More papers cited, and a minor reorganisation of the last
section, since last version. Typo corrected in the statement of Theorem
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