Permutation Statistics and Linear Extensions of Posets
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Trivial Meet and Join Within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected]
Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of 1-1-2014 Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Marquette University, [email protected] Adam Hammett Bethel College - Mishawaka Published version. Electronic Journal of Combinatorics, Vol. 21, No. 3 (2014). Permalink. © 2014 The Authors. Used with permission. Trivial Meet and Join within the Lattice of Monotone Triangles John Engbers Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI, U.S.A. [email protected] Adam Hammett Department of Mathematical Sciences Bethel College Mishawaka, IN, U.S.A. [email protected] Submitted: Jan 22, 2014; Accepted: Jul 10, 2014; Published: Jul 21, 2014 Mathematics Subject Classifications: 05A05, 05A16, 06A20 Abstract The lattice of monotone triangles (Mn; 6) ordered by entry-wise comparisons is studied. Let τmin denote the unique minimal element in this lattice, and τmax the unique maximum. The number of r-tuples of monotone triangles (τ1; : : : ; τr) with minimal infimum τmin (maximal supremum τmax, resp.) is shown to asymptotically r−1 approach rjMnj as n ! 1. Thus, with high probability this event implies that one of the τi is τmin (τmax, resp.). Higher-order error terms are also discussed. Keywords: monotone triangles; permutations; square ice; alternating sign matri- ces; meet; join 1 Introduction and statement -
2/3 Conjecture
PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics | Doctor of Philosophy 2017 ABSTRACT PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson Let (P; ≤) be a finite partially ordered set, also called a poset, and let n denote the cardinality of P . Fix a natural labeling on P so that the elements of P correspond to [n] = f1; 2; : : : ; ng. A linear extension is an order-preserving total order x1 ≺ x2 ≺ · · · ≺ xn on the elements of P , and more compactly, we can view this as the permutation x1x2 ··· xn in one-line notation. For distinct elements x; y 2 P , we define P(x ≺ y) to be the proportion 1 of linear extensions of P in which x comes before y. For 0 ≤ α ≤ 2, we say (x; y) is an α-balanced pair if α ≤ P(x ≺ y) ≤ 1 − α: The 1=3 − 2=3 Conjecture states that every finite partially ordered set that is not a chain has a 1=3-balanced pair. This dissertation focuses on showing the conjecture is true for certain types of partially ordered sets. We begin by discussing a special case, namely when a partial order is 1=2-balanced. For example, this happens when the poset has an automorphism with a cycle of length 2. We spend the remainder of the text proving the conjecture is true for some lattices, including Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. -
Counting Linear Extensions in Practice: MCMC Versus Exponential Monte Carlo
The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18) Counting Linear Extensions in Practice: MCMC versus Exponential Monte Carlo Topi Talvitie Kustaa Kangas Teppo Niinimaki¨ Mikko Koivisto Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science University of Helsinki Aalto University Aalto University University of Helsinki topi.talvitie@helsinki.fi juho-kustaa.kangas@aalto.fi teppo.niinimaki@aalto.fi mikko.koivisto@helsinki.fi Abstract diagram (Kangas et al. 2016), or the treewidth of the incom- parability graph (Eiben et al. 2016). Counting the linear extensions of a given partial order is a If an approximation with given probability is sufficient, #P-complete problem that arises in numerous applications. For polynomial-time approximation, several Markov chain the problem admits a polynomial-time solution for all posets Monte Carlo schemes have been proposed; however, little is using Markov chain Monte Carlo (MCMC) methods. The known of their efficiency in practice. This work presents an first fully-polynomial time approximation scheme of this empirical evaluation of the state-of-the-art schemes and in- kind for was based on algorithms for approximating the vol- vestigates a number of ideas to enhance their performance. In umes of convex bodies (Dyer, Frieze, and Kannan 1991). addition, we introduce a novel approximation scheme, adap- The later improvements were based on rapidly mixing tive relaxation Monte Carlo (ARMC), that leverages exact Markov chains in the set of linear extensions (Karzanov exponential-time counting algorithms. We show that approx- and Khachiyan 1991; Bubley and Dyer 1999) combined imate counting is feasible up to a few hundred elements on with Monte Carlo counting schemes (Brightwell and Win- various classes of partial orders, and within this range ARMC kler 1991; Banks et al. -
On Comparability of Random Permutations
ON COMPARABILITY OF RANDOM PERMUTATIONS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Adam Hammett, B.S. ***** The Ohio State University 2007 Dissertation Committee: Approved by Dr. Boris Pittel, Advisor Dr. Gerald Edgar Advisor Dr. Akos Seress Graduate Program in Mathematics ABSTRACT Two permutations of [n] := {1, 2, . , n} are comparable in the Bruhat order if one can be obtained from the other by a sequence of transpositions decreasing the number of inversions. We show that the total number of pairs of permutations (π, σ) with π ≤ σ is of order (n!)2/n2 at most. Equivalently, if π, σ are chosen uniformly at random and independently of each other, then P (π ≤ σ) is of order n−2 at most. By a direct probabilistic argument we prove P (π ≤ σ) is of order (0.708)n at least, so that there is currently a wide qualitative gap between the upper and lower bounds. Next, emboldened by a connection with Ferrers diagrams and plane partitions implicit in Bressoud’s book [13], we return to the Bruhat order upper bound and show that for n-permutations π1, . , πr selected independently and uniformly at random, −r(r−1) P (π1 ≤ · · · ≤ πr) = O n , thus providing an extension of our result for pairs of permutations to chains of length r > 2. Turning to the related weak order “” – when only adjacent transpositions are ∗ admissible – we use a non-inversion set criterion to prove that Pn := P (π σ) is pn ∗ submultiplicative, thus showing existence of ρ = lim Pn . -
A Selection of Maximal Elements Under Non-Transitive Indifferences
Munich Personal RePEc Archive A selection of maximal elements under non-transitive indifferences Alcantud, José Carlos R. and Bosi, Gianni and Zuanon, Magalì Universidad de Salamanca, University of Trieste, University of Brescia 4 August 2009 Online at https://mpra.ub.uni-muenchen.de/16601/ MPRA Paper No. 16601, posted 10 Aug 2009 08:03 UTC A selection of maximal elements under non-transitive indifferences Jos´eCarlos R. Alcantud ∗,1 Facultad de Econom´ıay Empresa, Universidad de Salamanca, E 37008 Salamanca, Spain Gianni Bosi Facolt`adi Economia, Universit`adegli Studi di Trieste, Piazzale Europa 1, 34127 Trieste, Italy Magal`ıZuanon Facolt`adi Economia, Universit`adegli Studi di Brescia, Contrada Santa Chiara 50, 25122 Brescia, Italy Abstract In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of “undominated maximals” (cf., Peris and Subiza [7]). Provided that an agent’s binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alterna- tives is finite. In the case of semiorders, proceeding in this way is the same as using Luce’s selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known “sugar example” by Luce. Key words: Maximal element, Selection of maximals, Acyclicity, Interval order, Semiorder JEL Classification: D11. Preprint submitted to Elsevier August 4, 2009 1 Introduction Even though there are arguments to ensure the existence of maximal elements for binary relations in very general settings, this concept does not always explain choice under non-transitive indifference well. -
Generalized Quotients in Coxeter Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 308, Number 1, July 1988 GENERALIZED QUOTIENTS IN COXETER GROUPS ANDERS BJORNER AND MICHELLE L. WACHS ABSTRACT. For (W, S) a Coxeter group, we study sets of the form W/V = {w e W | l{wv) = l(w) + l(v) for all v e V}, where V C W. Such sets W/V, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when VCS (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in W/V, for general V C W, are lexicographically shellable. The Mobius function on W/V under Bruhat order takes values in { —1,0, +1}. For finite groups W, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order. Descent classes Dj = {w e W \ l{ws) < l(w) o s 6 I, for all s G S}, I C S, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. -
Counting Linear Extensions of Restricted Posets
COUNTING LINEAR EXTENSIONS OF RESTRICTED POSETS SAMUEL DITTMER? AND IGOR PAK? Abstract. The classical 1991 result by Brightwell and Winkler [BW91] states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for posets of height two is #P-complete. Furthermore, we prove that this holds for incidence posets of graphs. Finally, we prove that the number of linear extensions for posets of dimension two is #P-complete. 1. Introduction Counting linear extensions (#LE) of a finite poset is a fundamental problem in both Combinatorics and Computer Science, with connections and applications ranging from Sta- tistics to Optimization, to Social Choice Theory. It is primarily motivated by the following basic question: given a partial information of preferences between various objects, what are the chances of other comparisons? In 1991, Brightwell and Winkler showed that #LE is #P-complete [BW91], but for var- ious restricted classes of posets the problem remains unresolved. Notably, they conjectured that the following problem is #P-complete: #H2LE (Number of linear extensions of height-2 posets) Input: A partially ordered set P of height 2. Output: The number e(P ) of linear extensions. Here height two means that P has two levels, i.e. no chains of length 3. This problem has been open for 27 years, most recently reiterated in [Hub14, LS17]. Its solution is the first result in this paper. Theorem 1.1. #H2LE is #P-complete. Our second result is an extension of Theorem 1.1. -
Mathematical Supplement
Appendix A Mathematical Supplement A.1 Elements of the Theory of Binary Relations A.1.1 Basic Operations on Sets Let us recall the definitions of some operations that are often performed on sets, and define the sets resulting from these operations. The union (or sum) of two sets A and B is the set [ A B consisting of all elements that belong to at least one of the sets A and B. The union of any number of sets is defined analogously: if we are given an arbitrary indexed family of sets, that is, a collection of sets Aa in which the index a runs through an arbitrary set (the index set), the union of the sets Aa is denoted by [ Aa : a By definition, this union consists of all elements that belong to at least one of the sets Aa . In addition, we agree that if the domain of the index a is not indicated, then the union extends over all the values that a may take in the given problem. The same refers to the operation of intersection. The intersection of two sets A and B is the set \ A B that consists of all elements belonging to both A and B. Analogously, the inter- section of an arbitrary (finite or infinite) number of sets Aa is the set \ Aa a 209 210 A Mathematical Supplement of all elements that belong to all of the sets Aa . The difference of the sets A and B is the set A n B consisting of those elements in A that are not contained in B. -
Quasitrivial Semigroups: Characterizations and Enumerations Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal
Quasitrivial semigroups: Characterizations and enumerations Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal To cite this version: Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal. Quasitrivial semigroups: Characterizations and enumerations. Semigroup Forum, Springer Verlag, 2019, 98 (3), pp.472-498. 10.1007/s00233- 018-9928-3. hal-01826868v2 HAL Id: hal-01826868 https://hal.inria.fr/hal-01826868v2 Submitted on 2 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASITRIVIAL SEMIGROUPS: CHARACTERIZATIONS AND ENUMERATIONS MIGUEL COUCEIRO, JIMMY DEVILLET, AND JEAN-LUC MARICHAL ABSTRACT. We investigate the class of quasitrivial semigroups and provide various char- acterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. 1. INTRODUCTION Let X be an arbitrary nonempty set. We use the symbol Xn if X contains n ≥ 1 elements, in which case we assume w.l.o.g. that Xn = {1; : : : ; n}. In this paper we investigate the class of binary operations F ∶ X2 → X that are asso- ciative and quasitrivial, where quasitriviality means that F always outputs one of its input values. -
Choice, Extension, Conservation. from Transfinite to finite Methods in Abstract Algebra
Choice, extension, conservation. From transfinite to finite methods in abstract algebra Daniel Wessel Universita` degli Studi di Trento Universita` degli Studi di Verona December 3, 2017 Doctoral thesis in Mathematics Joint doctoral programme in Mathematics, 30th cycle Department of Mathematics, University of Trento Department of Computer Science, University of Verona Academic year 2017/18 Supervisor: Peter Schuster, University of Verona Trento, Italy December 3, 2017 Abstract Maximality principles such as the ones going back to Kuratowski and Zorn ensure the existence of higher type ideal objects the use of which is commonly held indispensable for mathematical practice. The modern turn towards computational methods, which can be witnessed to have a strong influence on contemporary foundational studies, encourages a reassessment within a constructive framework of the methodological intricacies that go along with invocations of maximality principles. The common thread that can be followed through the chapters of this thesis is explained by the attempt to put the widespread use of ideal objects under constructive scrutiny. It thus walks the tracks of a revised Hilbert's programme which has inspired a reapproach to constructive algebra by finitary means, and for which Scott's entailment relations have already shown to provide a vital and utmost versatile tool. In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory; the notion of Jacobson radical is brought from com- mutative -
Arxiv:1604.04772V2 [Cs.DC] 28 Apr 2016
Abstract Graph Machine Thejaka Amila Kanewala Marcin Zalewski Andrew Lumsdaine Center for Research in Extreme Scale Technologies (CREST) Indiana University, IN, USA {thejkane,zalewski,lums}@indiana.edu ABSTRACT Graph Algorithms An Abstract Graph Machine(AGM) is an abstract model for dis- tributed memory parallel stabilizing graph algorithms. A stabilizing Set operations Nested Parallel Data Driven Iterative based algorithm starts from a particular initial state and goes through series Algorithms of different state changes until it converges. The AGM adds work de- Algorithms pendency to the stabilizing algorithm. The work is processed within the processing function. All processes in the system execute the Figure 1: Graph algorithm classification based on their solution same processing function. Before feeding work into the processing approach. function, work is ordered using a strict weak ordering relation. The (MST). While Bor˚uvka’s Minimum Spanning Tree (MST) uses set strict weak ordering relation divides work into equivalence classes, operations in its solution, the Dijkstra’s Single Source Shortest Path hence work within a single equivalence class can be processed in (SSSP) algorithm relies on priority based ordering of distances of parallel, but work in different equivalence classes must be executed neighbors to calculate the SSSP. Therefore, the solution approach in the order they appear in equivalence classes. The paper presents used in Dijkstra’s SSSP is different from the solution approach used the AGM model, semantics and AGM models for several existing in Bor˚uvka’s MST algorithm. distributed memory parallel graph algorithms. Based on the solution approach, we classify parallel graph algo- rithms into four categories (Figure 1): 1. -
TM-808C: Preferences, Risk Neutrality, and Utility Functions
Preferences, Risk Neutrality, and Utility Functions James C. Alexander Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7058 Matthew J. Sobel ∗ Department of Operations, Weatherhead School of Management, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7235, 216-368-6003 Abstract A binary preference relation on a real vector space V with zero element 0 sat- isfying four (natural) axioms induces a utility function U = f ◦ u : V ! R where u: V ! R is linear as a map of vector spaces and f : R ! R is weakly monotone. The weak ordering and non-triviality axioms are familiar, but the continuity ax- iom does not require topological assumptions. The fourth axiom (decomposition) is X − Y () 0 ) X () Y (X, Y 2 V ). The utility function U : R ! R can be taken to be linear if and only if also satisfies the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero pro- cess, it is known that the four axioms imply the existence of discount factors and, with the converse, the linearity of an intra-period utility function. Thus preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied. One consequence is that, within the developed context, a utility function that is not risk neutral can be replaced by one that is risk neutral and still be consistent with the underlying preference relation. Previous axiomati- zations of the discounted utility model by T. C.