Counting linear extension majority cycles in partially ordered sets on up to 13 elements

It is well known that the linear extension majority relation of a partially ordered set (P,@?"P) can contain cycles when at least 9 elements are present in P. Computer experiments have uncovered all posets with 9 elements containing such cycles and limited frequency estimates for linear extension majority cycles (or LEM cycles) in posets on up to 12 elements are available. In this contribution, we present an efficient approach which allows us to count and store all posets containing LEM cycles on up to 13 elements.

[1]  B. De Baets,et al.  On the random generation and counting of weak order extensions of a poset with given class cardinalities , 2007, Inf. Sci..

[2]  Peter C. Fishburn,et al.  A comparative analysis of methods for constructing weak orders from partial orders , 1975 .

[3]  Peter C. Fishburn,et al.  Linear extension majority cycles in height-1 orders , 1990 .

[4]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[5]  Jeff Kahn,et al.  Log-Concave Functions And Poset Probabilities , 1998, Comb..

[6]  Rainer Brüggemann,et al.  Basic principles of Hasse diagram technique in chemistry. , 2008, Combinatorial chemistry & high throughput screening.

[7]  William V. Gehrlein,et al.  The effectiveness of weighted scoring rules when pairwise majority rule cycles exist , 2004, Math. Soc. Sci..

[8]  W. V. Gehrlein.  Frequency estimates for linear extension majority cycles on partial orders , 1991 .

[9]  Bernhard Ganter,et al.  On linear extensions of ordered sets with a symmetry , 1987, Discret. Math..

[10]  Peter C. Fishburn,et al.  Linear extension majority cycles on partial orders , 1990 .

[11]  Peter B. Sørensen,et al.  Improving the Predicting Power of Partial Order Based QSARs through Linear Extensions , 2002, J. Chem. Inf. Comput. Sci..

[12]  Peter C. Fishburn.  Proportional transitivity in linear extensions of ordered sets , 1986, J. Comb. Theory, Ser. B.

[13]  Bernard De Baets,et al.  On the cycle-transitivity of the mutual rank probability relation of a poset , 2010, Fuzzy Sets Syst..

[14]  Brendan D. McKay,et al.  Posets on up to 16 Points , 2002, Order.

[15]  Stefan Felsner,et al.  Balancing pairs and the cross product conjecture , 1995 .

[16]  P. Sørensen,et al.  Evaluation of the ranking probabilities for partial orders based on random linear extensions. , 2003, Chemosphere.

[17]  Rainer Brüggemann,et al.  Estimation of Averaged Ranks by a Local Partial Order Model , 2004, J. Chem. Inf. Model..

[18]  Bernard De Baets,et al.  Graded Stochastic Dominance as a Tool for Ranking the Elements of a Poset , 2006, SMPS.

[19]  Rainer Brüggemann,et al.  A hitchhiker's guide to poset ranking. , 2008, Combinatorial chemistry & high throughput screening.

[20]  Lhouari Nourine,et al.  Efficient algorithms on distributive lattices , 2001, Discret. Appl. Math..

[21]  Rainer Brüggemann,et al.  Improved Estimation of the Ranking Probabilities in Partial Orders Using Random Linear Extensions by Approximation of the Mutual Ranking Probability , 2003, J. Chem. Inf. Comput. Sci..

[22]  G. P. Patil,et al.  Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization , 2004, Environmental and Ecological Statistics.

[23]  PETER C. FISHBURN.  On linear extension majority graphs of partial orders , 1976, J. Comb. Theory, Ser. B.

[24]  Peter C. Fishburn,et al.  Linear extension majority cycles for small (n⩽9) partial orders , 1990 .

[25]  Yang Yu.  On Proportional Transitivity of Ordered Sets , 1998 .

[26]  Graham R. Brightwell,et al.  Balanced pairs in partial orders , 1999, Discret. Math..

[27]  Peter C. Fishburn.  On the family of linear extensions of a partial order , 1974 .

[28]  Doron Rotem,et al.  An Algorithm to Generate all Topological Sorting Arrangements , 1981, Computer/law journal.

[29]  Rajkumar Roy,et al.  Advances in Soft Computing , 2018, Lecture Notes in Computer Science.

[30]  Martin Aigner.  Combinatorial search , 1988 .

[31]  Bernard De Baets,et al.  Exploiting the Lattice of Ideals Representation of a Poset , 2006, Fundam. Informaticae.