45 research outputs found
On the Factorization of Graphs with Exactly One Vertex of Infinite Degree
AbstractWe give a necessary and sufficient condition for the existence of a 1-factor in graphs with exactly one vertex of infinite degree
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
Multivariate Fuss-Catalan numbers
Catalan numbers enumerate binary trees and
Dyck paths. The distribution of paths with respect to their number of
factors is given by ballot numbers .
These integers are known to satisfy simple recurrence, which may be visualised
in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is
surprising that the extension of this construction to 3 dimensions generates
integers that give a 2-parameter distribution of , which may be called order-3 Fuss-Catalan numbers, and
enumerate ternary trees. The aim of this paper is a study of these integers
. We obtain an explicit formula and a description in terms of trees
and paths. Finally, we extend our construction to -dimensional arrays, and
in this case we obtain a -parameter distribution of , the number of -ary trees
The Bohnenblust–Spitzer Algorithm and its Applications
The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, variances, and concentration inequalities for several random variables associated with a random walk including the number of vertices and length of the convex minorant, concave majorant, and convex hull
Combinatorics of diagonally convex directed polyominoes
AbstractA new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney's generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q-enumeration of this object can be solved by an application of Gessel's q-analog of the Lagrange inversion formula
The Distribution of Patterns in Random Trees
Let denote the set of unrooted labeled trees of size and let
be a particular (finite, unlabeled) tree. Assuming that every tree of
is equally likely, it is shown that the limiting distribution as
goes to infinity of the number of occurrences of as an induced subtree is
asymptotically normal with mean value and variance asymptotically equivalent to
and , respectively, where the constants and
are computable