1,213 research outputs found
Some aspects of (r,k)-parking functions
An \emph{-parking function} of length may be defined as a sequence
of positive integers whose increasing rearrangement
satisfies . The case
corresponds to ordinary parking functions. We develop numerous properties of
-parking functions. In particular, if denotes the
Frobenius characteristic of the action of the symmetric group
on the set of all -parking functions of length , then we find a
combinatorial interpretation of the coefficients of the power series for any . When ,
this power series is just ; when , we
obtain a dual to -parking functions. We also give a -analogue of this
result. For fixed , we can use the symmetric functions to
define a multiplicative basis for the ring of symmetric functions. We
investigate some of the properties of this basis
From Fibonacci Numbers to Central Limit Type Theorems
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers
. Lekkerkerker proved that the average number of
summands for integers in is , with the
golden mean. This has been generalized to the following: given nonnegative
integers with and recursive sequence
with , and
, every positive
integer can be written uniquely as under natural constraints on
the 's, the mean and the variance of the numbers of summands for integers
in are of size , and the distribution of the numbers of
summands converges to a Gaussian as goes to the infinity. Previous
approaches used number theory or ergodic theory. We convert the problem to a
combinatorial one. In addition to re-deriving these results, our method
generalizes to a multitude of other problems (in the sequel paper \cite{BM} we
show how this perspective allows us to determine the distribution of gaps
between summands in decompositions). For example, it is known that every
integer can be written uniquely as a sum of the 's, such that every
two terms of the same (opposite) sign differ in index by at least 4 (3). The
presence of negative summands introduces complications and features not seen in
previous problems. We prove that the distribution of the numbers of positive
and negative summands converges to a bivariate normal with computable, negative
correlation, namely .Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the
number of summands in Zeckendorf decompositions. Version 2.0 (mostly
correcting missing references to previous literature
Performance Guarantees for Distributed Reachability Queries
In the real world a graph is often fragmented and distributed across
different sites. This highlights the need for evaluating queries on distributed
graphs. This paper proposes distributed evaluation algorithms for three classes
of queries: reachability for determining whether one node can reach another,
bounded reachability for deciding whether there exists a path of a bounded
length between a pair of nodes, and regular reachability for checking whether
there exists a path connecting two nodes such that the node labels on the path
form a string in a given regular expression. We develop these algorithms based
on partial evaluation, to explore parallel computation. When evaluating a query
Q on a distributed graph G, we show that these algorithms possess the following
performance guarantees, no matter how G is fragmented and distributed: (1) each
site is visited only once; (2) the total network traffic is determined by the
size of Q and the fragmentation of G, independent of the size of G; and (3) the
response time is decided by the largest fragment of G rather than the entire G.
In addition, we show that these algorithms can be readily implemented in the
MapReduce framework. Using synthetic and real-life data, we experimentally
verify that these algorithms are scalable on large graphs, regardless of how
the graphs are distributed.Comment: VLDB201
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