33 research outputs found
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for equipped with a
Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond
graph equipped with uniform measure and Hamming distance. We also prove that in
general probability metric spaces the in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
Hereditary chip-firing models generalize the Abelian sandpile model and the
cluster firing model to an exponential family of games induced by covers of the
vertex set. This generalization retains some desirable properties, e.g.
stabilization is independent of firings chosen and each chip-firing equivalence
class contains a unique recurrent configuration. In this paper we present an
explicit bijection between the recurrent configurations of a hereditary
chip-firing model on a graph and its spanning trees.Comment: 13 page
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur
The structure of the consecutive pattern poset
The consecutive pattern poset is the infinite partially ordered set of all
permutations where if has a subsequence of adjacent
entries in the same relative order as the entries of . We study the
structure of the intervals in this poset from topological, poset-theoretic, and
enumerative perspectives. In particular, we prove that all intervals are
rank-unimodal and strongly Sperner, and we characterize disconnected and
shellable intervals. We also show that most intervals are not shellable and
have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR