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### The Average Number of Linear Extensions of a Partial Order

G Brightwell
1996 Journal of combinatorial theory. Series A
This gives us an asymptotic formula for (a) the average number of linear extensions of an n-element partial order and (b) the number of suborders of an n-element linear order.  ...  We give a shorter proof of their result and extend it to count the number of pairs (P, -<), where P is a partial order on [hi and -< is a linear extension of P.  ...  ACKNOWLEDGMENT We thank Jim Fill for his explanation of the similarity between the numerical values of ~'1 and ~2 and Ivan Rival for drawing our attention to the work of Avann.  ...

### The average number of linear extensions of a partial order

Graham Brightwell, Hans Jürgen Prömel, Angelika Steger
1996 Journal of combinatorial theory. Series A
This gives us an asymptotic formula for (a) the average number of linear extensions of an n-element partial order and (b) the number of suborders of an n-element linear order.  ...  We give a shorter proof of their result and extend it to count the number of pairs (P, -<), where P is a partial order on [hi and -< is a linear extension of P.  ...  ACKNOWLEDGMENT We thank Jim Fill for his explanation of the similarity between the numerical values of ~'1 and ~2 and Ivan Rival for drawing our attention to the work of Avann.  ...

### Balanced pairs in partial orders

Graham Brightwell
1999 Discrete Mathematics
An c~-balanced pair in a partially ordered set P = (X, <) is a pair (x, y) of elements of X such that the proportion of linear extensions of P with x below y lies between ~ and 1 -~.  ...  The 1/3-2/3 Conjecture states that, in every finite partial order P, not a chain, there is a 1/3-balanced pair. This was first conjectured in a 1968 paper of Kislitsyn, and remains unsolved.  ...  The number of linear orders consistent with the information < is simply the number e(P) of linear extensions of the partially ordered set P = (X, < ).  ...

### Belief propagation for permutations, rankings, and partial orders [article]

George T. Cantwell, Cristopher Moore
2022 arXiv   pre-print
In addition, the Bethe free energy lets us approximate the number of linear extensions of a partial order and perform model selection between competing probabilistic models, such as the Bradley-Terry-Luce  ...  Many datasets give partial information about an ordering or ranking by indicating which team won a game, which item a user prefers, or who infected whom.  ...  A linear extension of a partial order is a total order that satisfies all the constraints of the partial order; it is is equivalent to a topological ordering of the corresponding directed acyclic graph  ...

### Computational Complexity of a Solution for Directed Graph Cooperative Games

Ayumi Igarashi, Yoshitsugu Yamamoto
2013 Journal of the Operations Research Society of China
We prove this by a reduction from counting the number of all linear extensions of a partial order, which has been shown by Brightwell et al. to be a #P -complete counting problem.  ...  The average covering tree value is defined as the average of marginal contribution vectors corresponding to the specific set of rooted trees, and coincides with the Shapley value when the game has complete  ...  Acknowledgements We wish to thank the two anonymous reviewers for their constructive suggestions and comments. The comments have helped us significantly improve the paper.  ...

### Pseudo Boolean Programming for Partially Ordered Genomes [chapter]

Sébastien Angibaud, Guillaume Fertin, Annelyse Thévenin, Stéphane Vialette
2009 Lecture Notes in Computer Science
Next, we propose an algorithm to find linear extensions of two partial orders that maximize the number of adjacencies.  ...  Starting from a partial order between genes of a genome, one method to find a total order consists in optimizing a given measure between a linear extension of this partial order and a given total order  ...  Fig. 3 . 3 Program Adj-1PO computes a linear extension of a partially ordered genome that maximizes the number of adjacencies between this linear extension and the identity.  ...

### EXPLOITING ORDINAL DATA FOR SUBJECTIVE WELL-BEING EVALUATION

Marco Fattore, Filomena Maggino, Alberto Arcagni
2015 Statistics in Transition New Series
In this paper we take a different path and prove that the evaluation of multidimensional ordinal well-being can be addressed in an effective and consistent way, using the theory of partially ordered sets  ...  We first show that the proper evaluation space of well-being is the partially ordered set of achievement profiles and that its structure depends upon the importance assigned to well-being attributes.  ...  An extension λ which is also a linear order is called a linear extension of Π. The set of linear extensions of Π is denoted by Ω(Π).  ...

### Learning Plackett-Luce Mixtures from Partial Preferences

Ao Liu, Zhibing Zhao, Chao Liao, Pinyan Lu, Lirong Xia
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
The core of our framework is the efficient sampling of linear extensions of partial orders under Plackett-Luce model.  ...  Experiments on real-world data show that the likelihood of test dataset increases when (i) partial orders provide more information; or (ii) the number of components in mixtures of PlackettLuce model increases  ...  Let Ext(V j ) be the set of all linear extensions of V j , which consists of all linear orders consistent with V j . A linear extension bridges a partial order (observed data) and a linear order.  ...

### The Decision-Theoretic Interactive Video Advisor [article]

2013 arXiv   pre-print
It uses a novel similarity measure based on the concept of the probability of conflict between two orderings of items.  ...  The need to help people choose among large numbers of items and to filter through large amounts of information has led to a flood of research in construction of personal recommendation agents.  ...  The consistent total orders extension, the algorithm involves running a Monte Carlo of a partial order are called linear extensions.  ...

### Average height in a partially ordered set

P. Winkler
1982 Discrete Mathematics
The average height of an element x in a finite poset P is the expected number of elements below x in a random linear extension of P.  ...  We prove a number of theorems about average height, some intuitive and some not, using a recent result of L.A. Shepp.  ...  (v) is a partial ordering which extends the ordering on P, and is often linear, 1'9~s providing a canonical linear extension.  ...

### The geometry of partial fitness orders and an efficient method for detecting genetic interactions

Caitlin Lienkaemper, Lisa Lamberti, James Drain, Niko Beerenwinkel, Alex Gavryushkin
2018 Journal of Mathematical Biology
to their fitness, i.e. from partial fitness orders.  ...  We present an efficient computational approach for detecting genetic interactions from fitness comparison data together with a geometric interpretation using polyhedral cones associated to partial orderings  ...  AG was partially supported by Royal Society of New Zealand through Rutherford Discovery Fellowship, contract RDF-UOO1702.  ...

### Ranking with Uncertain Scores

Mohamed A. Soliman, Ihab F. Ilyas
2009 Proceedings / International Conference on Data Engineering
We describe and analyze a set of efficient query evaluation algorithms. We show that our techniques can be used to solve the problem of rank aggregation in partial orders.  ...  In this paper, we present a new probabilistic model, based on partial orders, to encapsulate the space of possible rankings originating from score uncertainty.  ...  The linear extensions of a partial order are all possible topological sorts of the partial order graph (i.e., the relative order of any two elements in any linear extension does not violate the set of  ...

### Page 4827 of Mathematical Reviews Vol. , Issue 83m [page]

1983 Mathematical Reviews
Average height in a partially ordered set. Discrete Math. 39 (1982), no. 3, 337-341.  ...  For example, let P be an n-element ordered set, let ®(P) be the family of all order-preserving bijections of P to {0,1,---,n—1}, and let L( P) denote the cardinality of ®( P), that is, the number of linear  ...

### The Mixing of Markov Chains on Linear Extensions in Practice

Topi Talvitie, Teppo Niinimäki, Mikko Koivisto
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence
We investigate almost uniform sampling from the set of linear extensions of a given partial order.  ...  We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders.  ...  Acknowledgements This work was supported in part by the Academy of Finland, under Grant 276864 and Grant 303816.  ...

### Estimating Octanol / Water Partition Coefficients by Order Preserving Mappings

Rainer Bruggemann, Guillermo Restrepo
2013 Croatica Chemica Acta
Douglas Jay Klein and other researchers have seen that there is a great potential in applying partial order in the field of QSAR.  ...  It turned out that the application of average heights, h av , as predictors of a linear log K OW estimation leads to reasonable results in the application example of chlorophenols. (  ...  The sequence of edge labels is a linear extension of (X, IB), for example c  d  a  b is a linear extension of the Fence(4).  ...
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