1,182 research outputs found
Combinatorial proofs of some properties of tangent and Genocchi numbers
The tangent number is equal to the number of increasing labelled
complete binary trees with vertices. This combinatorial interpretation
immediately proves that is divisible by . However, a stronger
divisibility property is known in the studies of Bernoulli and Genocchi
numbers, namely, the divisibility of by . The
traditional proofs of this fact need significant calculations. In the present
paper, we provide a combinatorial proof of the latter divisibility by using the
hook length formula for trees. Furthermore, our method is extended to -ary
trees, leading to a new generalization of the Genocchi numbers
An edge-weighted hook formula for labelled trees
A number of hook formulas and hook summation formulas have previously
appeared, involving various classes of trees. One of these classes of trees is
rooted trees with labelled vertices, in which the labels increase along every
chain from the root vertex to a leaf. In this paper we give a new hook
summation formula for these (unordered increasing) trees, by introducing a new
set of indeterminates indexed by pairs of vertices, that we call edge weights.
This new result generalizes a previous result by F\'eray and Goulden, that
arose in the context of representations of the symmetric group via the study of
Kerov's character polynomials. Our proof is by means of a combinatorial
bijection that is a generalization of the Pr\"ufer code for labelled trees.Comment: 25 pages, 9 figures. Author-produced copy of the article to appear in
Journal of Combinatorics, including referee's suggestion
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
- …