scholarly journals Counting Set Systems by Weight

10.37236/1908 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Martin Klazar
Keyword(s):  
Set Partitions ◽  
Unique Root ◽  
Set Systems ◽  
Bell Number ◽  
Vertex Degrees ◽  
Sparse Set

Applying enumeration of sparse set partitions, we show that the number of set systems $H\subset\exp(\{1,2,\dots,n\})$ such that $\emptyset\notin H$, $\sum_{E\in H} |E|=n$ and $\bigcup_{E\in H} E=\{1,2,\dots,m\}$, $m\le n$, equals $(1/\log(2)+o(1))^nb_n$ where $b_n$ is the $n$-th Bell number. The same asymptotics holds if $H$ may be a multiset. If the vertex degrees in $H$ are restricted to be at most $k$, the asymptotics is $(1/\alpha_k+o(1))^nb_n$ where $\alpha_k$ is the unique root of $\sum_{i=1}^k x^i/i!-1$ in $(0,1]$.

scholarly journals INFINITE COMBINATORICS PLAIN AND SIMPLE

2018 ◽  
Vol 83 (3) ◽  
pp. 1247-1281 ◽  
Author(s):  
DÁNIEL T. SOUKUP ◽  
LAJOS SOUKUP
Keyword(s):  
Chromatic Number ◽  
Set Theory ◽  
Wide Applicability ◽  
Elementary Submodels ◽  
Set Systems ◽  
Point Set ◽  
Infinite Combinatorics ◽  
Sparse Set ◽  
Paradoxical Decompositions ◽  
General Method

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

scholarly journals On Cross-Intersecting Families of Set Partitions

10.37236/2191 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Set Partitions ◽  
Large N ◽  
Bell Number ◽  
Intersecting Families

Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal A_1,\mathcal A_2\subseteq \mathcal{B}(n)$ are cross-intersecting i.e. for all $A_1\in \mathcal A_1$ and $A_2\in \mathcal A_2$, we have $A_1\cap A_2\neq\varnothing$. It is proved that for sufficiently large $n$,\[ \vert \mathcal A_1\vert\vert \mathcal A_2\vert\leq B_{n-1}^2\]where $B_{n}$ is the $n$-th Bell number. Moreover, equality holds if and only if $\mathcal{A}_1=\mathcal A_2$ and $\mathcal A_1$ consists of all set partitions with a fixed singleton.

scholarly journals Unavoidable subprojections in union-closed set systems of infinite breadth

2021 ◽  
Vol 94 ◽  
pp. 103311
Author(s):  
Yemon Choi ◽  
Mahya Ghandehari ◽  
Hung Le Pham
Keyword(s):  
Closed Set ◽  
Set Systems

scholarly journals The Fair Division of Hereditary Set Systems

2021 ◽  
Vol 9 (2) ◽  
pp. 1-19
Author(s):  
Z. Li ◽  
A. Vetta
Keyword(s):  
Recent Work ◽  
Polynomial Time ◽  
Simple Algorithm ◽  
Fair Division ◽  
Set Systems

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form a hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3666 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

scholarly journals Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Han-Young Kim
Keyword(s):  
Bell Polynomials ◽  
Bell Numbers ◽  
Bell Number ◽  
Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

scholarly journals The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

2021 ◽  
Author(s):  
Anne Driemel ◽  
André Nusser ◽  
Jeff M. Phillips ◽  
Ioannis Psarros
Keyword(s):  
Density Estimation ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Similarity Metrics ◽  
Vc Dimension ◽  
Set Systems ◽  
Polygonal Curves ◽  
The Individual ◽  
Curve Similarity ◽  
Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

scholarly journals Non-Homogeneous Markov Set Systems

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 471
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Asymptotic Behavior ◽  
Population Structure ◽  
Confidence Intervals ◽  
Limit Set ◽  
Stroke Patients ◽  
Relative Population ◽  
Set Systems ◽  
Population Structures ◽  
Basic Parameters ◽  
Initial Distributions

A more realistic way to describe a model is the use of intervals which contain the required values of the parameters. In practice we estimate the parameters from a set of data and it is natural that they will be in confidence intervals. In the present study, we study Non-Homogeneous Markov Systems (NHMS) processes for which the required basic parameters are in intervals. We call such processes Non-Homogeneous Markov Set Systems (NHMSS). First we study the set of the relative expected population structure of memberships and we prove that under certain conditions of convexity of the intervals of the parameters the set is compact and convex. Next, we establish that if the NHMSS starts with two different initial distributions sets and allocation probability sets under certain conditions, asymptotically the two expected relative population structures coincide geometrically fast. We continue proving a series of theorems on the asymptotic behavior of the expected relative population structure of a NHMSS and the properties of their limit set. Finally, we present an application for geriatric and stroke patients in a hospital and through it we solve problems that surface in an application.

Lower bound on the profile of degree pairs in cross-intersecting set systems

COMBINATORICA ◽  
2007 ◽  
Vol 27 (3) ◽  
pp. 399-405
Author(s):  
Zsuzsanna Szaniszló ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Set Systems

scholarly journals Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

2010 ◽  
Vol 224 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Set Systems

scholarly journals Structure and stability of triangle-free set systems

2006 ◽  
Vol 359 (1) ◽  
pp. 275-291 ◽  
Author(s):  
Dhruv Mubayi
Keyword(s):  
Structure And Stability ◽  
Set Systems ◽  
Free Set
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