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The Average Number of Linear Extensions of a Partial Order

1996
*
Journal of combinatorial theory. Series A
*

This gives us an asymptotic formula for (

doi:10.1006/jcta.1996.0017
fatcat:dpbhc4kebzhdxiiox36neuxcze
*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

1996
*
Journal of combinatorial theory. Series A
*

This gives us an asymptotic formula for (

doi:10.1016/s0097-3165(96)80001-x
fatcat:wzo6ajt2zzcntjqu4rn7emwbhi
*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

1999
*
Discrete Mathematics
*

An c~-balanced pair in

doi:10.1016/s0012-365x(98)00311-2
fatcat:6rg6a57emnbrdgmjgqga5br7k4
*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]

2022
*
arXiv
*
pre-print

In addition,

arXiv:2110.00513v2
fatcat:lq5nkrv3tvg4bjs2c23uueu52m
*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

2013
*
Journal of the Operations Research Society of China
*

We prove this by

doi:10.1007/s40305-013-0025-8
fatcat:yo7jlle3wne6xbsdcyyylmx2nu
*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]

2009
*
Lecture Notes in Computer Science
*

Next, we propose an algorithm to find

doi:10.1007/978-3-642-04744-2_11
fatcat:o7fasbqscnfx7oocw7hjymzv6i
*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

2015
*
Statistics in Transition New Series
*

In this paper we take

doi:10.21307/stattrans-2015-023
fatcat:xywqy6dul5agjlaqabhmndc27e
*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

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

arXiv:1301.6728v1
fatcat:kwm7cgfhv5a27mht4dyjaz7kt4
*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

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

2018
*
Journal of Mathematical Biology
*

to their fitness, i.e. from

doi:10.1007/s00285-018-1237-7
pmid:29736875
fatcat:swlr6dtq3vgrzmjgs7twgnkkve
*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

2009
*
Proceedings / International Conference on Data Engineering
*

We describe and analyze

doi:10.1109/icde.2009.102
dblp:conf/icde/SolimanI09
fatcat:4y62custw5hxlpp7gc2gvs3vzy
*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

2017
*
Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence
*

We investigate almost uniform sampling from

doi:10.24963/ijcai.2017/74
dblp:conf/ijcai/TalvitieNK17
fatcat:qmlxusczuzggnnyj5vrcuxv5pm
*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

2013
*
Croatica Chemica Acta
*

Douglas Jay Klein and other researchers have seen that there is

doi:10.5562/cca2296
fatcat:ybmscfr63becnm4ukdhc3t76m4
*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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