24 research outputs found
On Packing Densities of Set Partitions
We study packing densities for set partitions, which is a generalization of
packing words. We use results from the literature about packing densities for
permutations and words to provide packing densities for set partitions. These
results give us most of the packing densities for partitions of the set
. In the final section we determine the packing density of the set
partition .Comment: 12 pages, to appear in the Permutation Patterns edition of the
Australasian Journal of Combinatoric
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
Prolific Compositions
Under what circumstances might every extension of a combinatorial structure
contain more copies of another one than the original did? This property, which
we call prolificity, holds universally in some cases (e.g., finite linear
orders) and only trivially in others (e.g., permutations). Integer
compositions, or equivalently layered permutations, provide a middle ground. In
that setting, there are prolific compositions for a given pattern if and only
if that pattern begins and ends with 1. For each pattern, there is an easily
constructed automaton that recognises prolific compositions for that pattern.
Some instances where there is a unique minimal prolific composition for a
pattern are classified