scholarly journals On the Rank Function of a Differential Poset

10.37236/2258 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Richard P. Stanley ◽  
Fabrizio Zanello
Keyword(s):  
Asymptotic Formula ◽  
Projective Planes ◽  
Rank Function ◽  
Combinatorial Objects ◽  
Open Questions ◽  
Algebraic Properties ◽  
Interval Property ◽  
Differential Poset ◽  
Special Case ◽  
Young’S Lattice

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.

scholarly journals The Rank Function of a Positroid and Non-Crossing Partitions

10.37236/8256 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Mcalmon ◽  
Suho Oh
Keyword(s):  
Rank Function ◽  
Combinatorial Objects ◽  
Special Case

A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.

Weakly Isotopic Planar Ternary Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 32-36
Author(s):  
Frederick W. Stevenson
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Ternary Ring ◽  
Algebraic Relation ◽  
Planar Ternary Ring ◽  
Algebraic Properties

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

A Risk Model with Delayed Claims

2013 ◽  
Vol 50 (03) ◽  
pp. 686-702 ◽  
Author(s):  
Angelos Dassios ◽  
Hongbiao Zhao
Keyword(s):  
Asymptotic Formula ◽  
Poisson Process ◽  
Ruin Probability ◽  
Risk Model ◽  
Poisson Model ◽  
Exact Formula ◽  
Poisson Models ◽  
Asymptotic Expressions ◽  
Special Case

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

scholarly journals Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

10.37236/773 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jeremy F. Alm ◽  
Roger D. Maddux ◽  
Jacob Manske
Keyword(s):  
Complete Graph ◽  
Ramsey Theory ◽  
Projective Planes ◽  
Relation Algebras ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
Finite Set ◽  
Vertex Set ◽  
Special Case

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.

scholarly journals Permutations Avoiding Certain Partially-Ordered Patterns

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Kai Ting Keshia Yap ◽  
David Wehlau ◽  
Imed Zaguia
Keyword(s):  
Fibonacci Numbers ◽  
Recursive Algorithms ◽  
Relative Order ◽  
Partially Ordered ◽  
Combinatorial Objects ◽  
Open Questions

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev in 2019. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive these sets and their corresponding combinatorial objects in parallel, which yielded natural bijections. We also analysed an avoidance set whose simple permutations are enumerated by the Fibonacci numbers and derived an algorithm to obtain them recursively.

scholarly journals An Order on Circular Permutations

10.37236/9982 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Antoine Abram ◽  
Nathan Chapelier-Laget ◽  
Christophe Reutenauer
Keyword(s):  
Symmetric Group ◽  
Weak Order ◽  
Rank Function ◽  
Weyl Groups ◽  
Eulerian Numbers ◽  
Affine Weyl Groups ◽  
Special Formula ◽  
Circular Permutations ◽  
Semidistributive Lattice ◽  
Young’S Lattice

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

scholarly journals Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes

2018 ◽  
Vol 2020 (13) ◽  
pp. 3886-3901
Author(s):  
Michael Magee
Keyword(s):  
Asymptotic Formula ◽  
Projective Plane ◽  
Projective Planes ◽  
Hyperbolic Metric ◽  
Closed Geodesics ◽  
Real Projective Plane ◽  
Closed Curves ◽  
The Hyperbolic Metric ◽  
Simple Closed Geodesics

Abstract We prove that there is a true asymptotic formula for the number of one-sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic metric, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.

Erdős–Ko–Rado in Random Hypergraphs

2009 ◽  
Vol 18 (5) ◽  
pp. 629-646 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
TOM BOHMAN ◽  
DHRUV MUBAYI
Keyword(s):  
Maximum Size ◽  
Projective Planes ◽  
Uniform Hypergraph ◽  
Open Questions ◽  
Wide Range ◽  
Random Hypergraphs ◽  
The Way

Let 3 ≤k<n/2. We prove the analogue of the Erdős–Ko–Rado theorem for the randomk-uniform hypergraphGk(n,p) whenk< (n/2)1/3; that is, we show that with probability tending to 1 asn→ ∞, the maximum size of an intersecting subfamily ofGk(n,p) is the size of a maximum trivial family. The analogue of the Erdős–Ko–Rado theorem does not hold for allpwhenk≫n1/3.We give quite precise results fork<n1/2−ϵ. For largerkwe show that the random Erdős–Ko–Rado theorem holds as long aspis not too small, and fails to hold for a wide range of smaller values ofp. Along the way, we prove that every non-trivial intersectingk-uniform hypergraph can be covered byk2−k+ 1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.

Idempotent lifting and ring extensions

2016 ◽  
Vol 15 (06) ◽  
pp. 1650112 ◽  
Author(s):  
Alexander J. Diesl ◽  
Samuel J. Dittmer ◽  
Pace P. Nielsen
Keyword(s):  
Jacobson Radical ◽  
General Theorem ◽  
Positive Side ◽  
Open Questions ◽  
Morita Invariant ◽  
Ring Extensions ◽  
Special Case

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

scholarly journals A theory of spectral partitions of metric graphs

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo
Keyword(s):  
Spectral Gaps ◽  
Qualitative Properties ◽  
Metric Graphs ◽  
One Dimensional ◽  
Graph Partitions ◽  
Planar Domains ◽  
Open Questions ◽  
The One ◽  
Abstract Framework ◽  
Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

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