129 research outputs found
Telescoping Sums, Permutations, and First Occurrence Distributions
Telescoping sums very naturally lead to probability distributions on
. But are these distributions typically cosmetic and devoid of
motivation? In this paper we give three examples of "first occurrence"
distributions, each defined by telescoping sums, and that each arise from
concrete questions about the structure of permutations.Comment: 13 page
Universal Cycles of Restricted Words
A connected digraph in which the in-degree of any vertex equals its
out-degree is Eulerian, this baseline result is used as the basis of existence
proofs for universal cycles (also known as generalized deBruijn cycles or
U-cycles) of several combinatorial objects. We extend the body of known results
by presenting new results on the existence of universal cycles of monotone,
"augmented onto", and Lipschitz functions in addition to universal cycles of
certain types of lattice paths and random walks.Comment: 21 pages, 4 figure
Waiting Time Distribution for the Emergence of Superpatterns
Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking
values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of
X_1,...,X_t that contains, as an embedded subsequence, each of the
non-order-isomorphic subpatterns of length k. We focus on the non-trivial case
of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t
is a superpattern}Comment: 17 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
Maximum Number of Minimum Dominating and Minimum Total Dominating Sets
Given a connected graph with domination (or total domination) number
\gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of
dominating and total dominating sets of size \gamma. An exact answer is
provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page
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