727 research outputs found
On the non-existence of an R-labeling
We present a family of Eulerian posets which does not have any R-labeling.
The result uses a structure theorem for R-labelings of the butterfly poset.Comment: 6 pages, 1 figure. To appear in the journal Orde
On the structure of the space of geometric product-form models
This article deals with Markovian models defined on a finite-dimensional discrete state space and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object and explore its structure. We focus on models on an orthant [script Z]+n, which are homogeneous within subsets of [script Z]+n called walls, and permit only state transitions whose [parallel R: parallel] [parallel R: parallel][infty infinity]-length is 1. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. This principle holds also for state spaces with multiple corners (e.g., bounded boxes in [script Z]+n).\ud
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In addition, we consider models which are homogeneous throughout a finite-dimensional grid [script Z]n, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions, we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.\u
Cyclotomic factors of the descent set polynomial
We introduce the notion of the descent set polynomial as an alternative way
of encoding the sizes of descent classes of permutations. Descent set
polynomials exhibit interesting factorization patterns. We explore the question
of when particular cyclotomic factors divide these polynomials. As an instance
we deduce that the proportion of odd entries in the descent set statistics in
the symmetric group S_n only depends on the number on 1's in the binary
expansion of n. We observe similar properties for the signed descent set
statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up
notatio
Present Knowledge of the Systematics and Zoogeography of the Order Gorgonacea in Hawaii
Past knowledge of the order Gorgonacea in Hawaii is based almost
exclusively on the collections of the United States Fish Commission steamer Albatross
in 1902, which contain 52 species. Recent efforts to investigate the ecology of
precious coral have produced a new collection based on 183 dredge hauls and 10
dives with a submersible. This program is collectively referred to as the Sango
Expedition. Of 59 species of gorgonians obtained by the Sango Expedition, 13 are
considered to be new species and 28 new geographic records, bringing the total
number of species considered to be present in Hawaii to 93 species.
In contrast to the high diversity of gorgonians in the West Indies and the Indo-West-Pacific, the faunal list in Hawaii must still be considered depauperate. This is
especially true in shallow water <75 m), where only one species is known. Although climatic deterioration during the Pleistocene could account for the scarcity
of gorgonians in shallow water at the present time, this factor is unlikely to have
affected deeper species. Furthermore, one would expect to find a modern complement
of an ancestral faun a in shallow water if it had existed, as is true in the case of
reef corals. The paucity of gorgonians in Hawaii may be due to isolation, which
appears to have been a particularly effective barrier in shallow water. It is suggested
that the only accessible route to Hawaii for gorgonians has been in deep water
where, in the past, there were numerous stepping stones that may have aided dispersal.
Moreover, chemical and physical gradients in deep water are relatively low.
Why more deepwater species have not migrated into shallow water in Hawaii may
be a reflection of their stenotypic character
Classification of the factorial functions of Eulerian binomial and Sheffer posets
We give a complete classification of the factorial functions of Eulerian
binomial posets. The factorial function B(n) either coincides with , the
factorial function of the infinite Boolean algebra, or , the factorial
function of the infinite butterfly poset. We also classify the factorial
functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial
factorial function has Sheffer factorial function D(n) identical to
that of the infinite Boolean algebra, the infinite Boolean algebra with two new
coatoms inserted, or the infinite cubical poset. Moreover, we are able to
classify the Sheffer factorial functions of Eulerian Sheffer posets with
binomial factorial function as the doubling of an upside down
tree with ranks 1 and 2 modified.
When we impose the further condition that a given Eulerian binomial or
Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite
Boolean algebra or the infinite cubical lattice . We also
include several poset constructions that have the same factorial functions as
the infinite cubical poset, demonstrating that classifying Eulerian Sheffer
posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title
change. To appear in JCT
Level Eulerian Posets
The notion of level posets is introduced. This class of infinite posets has
the property that between every two adjacent ranks the same bipartite graph
occurs. When the adjacency matrix is indecomposable, we determine the length of
the longest interval one needs to check to verify Eulerianness. Furthermore, we
show that every level Eulerian poset associated to an indecomposable matrix has
even order. A condition for verifying shellability is introduced and is
automated using the algebra of walks. Applying the Skolem--Mahler--Lech
theorem, the -series of a level poset is shown to be a rational
generating function in the non-commutative variables and .
In the case the poset is also Eulerian, the analogous result holds for the
-series. Using coalgebraic techniques a method is developed to
recognize the -series matrix of a level Eulerian poset
The Tchebyshev transforms of the first and second kind
We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
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