DOCSLIB.ORG

The 1/3-2/3 Conjecture for Coxeter Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The 1/3-2/3 Conjecture for Coxeter Groups The 1/3-2/3 Conjecture for Coxeter groups Yibo Gao Joint work with: Christian Gaetz Massachusetts Institute of Technology UW Combinatorics and Geometry Seminar, 2020 Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 1 / 35 Overview 1 Linear extensions and the 1/3-2/3 Conjecture 2 Weak order and convex sets 3 The fully commutative case 4 Generalized semiorders 5 A uniform lower bound 6 More examples Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 2 / 35 Linear extensions of posets A linear extension of a poset P is an order-preserving bijection: λ : P ! [n] = f1; 2;:::; ng: Let E(P) be the set of linear extensions of P and e(P) be the number of linear extensions of P. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 3 / 35 The 1/3-2/3 Conjecture For p; q 2 P, let δ(p; q) be the fraction of linear extensions such that λ(p) < λ(q): jλ 2 E(P) j λ(p) < λ(q)j δ(p; q) = : e(P) Conjecture (Kislicyn 1968; Fredman 1974; Linial 1984) For any finite poset P that is not a total order, there exists p; q such that 1 2 ≤ δ(p; q) ≤ : 3 3 Define the balance constant to be b(P) = max minfδ(p; q); 1 − δ(p; q)g p;q2P Then the conjecture is saying that b(P) ≥ 1=3: Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 4 / 35 Why 1/3? Example p2 p1 p3 This poset has 3 linear extensions p1p2p3, p1p3p2, p3p1p2, with δ(p1; p2) = 1, δ(p1; p3) = 2=3, δ(p2; p3) = 1=3, thus b(P) = 1=3. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 5 / 35 Known Results It is nontrivial that b(P) is uniformly bounded away from zero: • b(P) ≥ 1=2e ≈ 0:1839 (Kahn and Linial 1991) • b(P) ≥ 3=11 ≈ 0:2727 (Kahn and Saks 1984) p • b(P) ≥ (5 − 5)=10 ≈ 0:2764 (Brightwell, Felsner, and Trotter 1995) The full conjecture is known for: • Width-2 posets • Semiorders (unit interval orders) • Height-2 posets • Series parallel posets (N-free posets) Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 6 / 35 Coxeter groups and the weak order A Coxeter group W is generated by a set of simple reflections S = fs1;:::; sr g such that 2 mi;j W = hs1;:::; sr j si = id; (si sj ) = id; 8i; ji for some mij 2 f2; 3;:::; 1g: Let `(w) denote the Coxeter length of w 2 W . The (left) weak order ≤L is generated by w l sw if `(w) + 1 = `(sw): Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 7 / 35 Coxeter groups and the weak order Let (W ; S) be a Coxeter system. The reflections T are the conjugates of simple reflections. The (right) inversion set of w 2 W is Inv(w) := ft 2 T j `(wt) < `(w)g: Proposition 0 0 0 For w; w 2 W , w ≤L w if and only if Inv(w) ⊂ Inv(w ): It's not hard to see that `(w) = jInv(w)j: Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 8 / 35 Convex sets in Coxeter groups Let H be the Hasse diagram of the weak order ≤L. A subset C ⊂ W is called convex if for any x; y 2 C, any shortest-length path between x and y remains in C. Proposition (Tits 1974) Any convex subset C ⊂ W is of the form B C = WA := fw 2 W j A ⊂ Inv(w) ⊂ Bg: For our purposes, we usually translate a convex subset C by some element w 2 W to assume that A = ;. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 9 / 35 Example: the symmetric group Sn The symmetric group Sn is a Coxeter group generated by S = f(12); (23);::: (n−1 n)g with reflections T = f(ij) j 1 ≤ i < j ≤ ng. The reflection (ij) is an inversion of w if w(i) > w(j). f(13);(23);(24)g C = f1234; 1243; 2134; 2143; 3142g is convex and C = W; . Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 10 / 35 Posets as convex sets in Sn For a poset P on fp1;:::; png, we identify E(P) as a set of permutations CP = fλ 2 E(P) j λ(1)λ(2) ··· λ(n) 2 Sng: Proposition The map P 7! CP gives a bijection between labeled poset on n elements and convex sets of Sn. Example p3 p4 p1 p2 corresponds to the convex set f1234; 2134; 2143; 1243; 2413g. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 11 / 35 Balance constants for Coxeter groups Recall the definition and we have jfλ 2 E(P) j λ(p ) < λ(p )gj jfw 2 C j (ij) 2 Inv(w)gj δ(p ; p ) = j i = P : j i e(P) e(P) This suggests the following definition. Let C ⊂ W be a convex set and t 2 T be an inversion. Define jfw 2 C j t 2 Inv(w)gj δ(t) = : jCj Define the balance constant to be b(C) = max minfδ(t); 1 − δ(t)g: t2T Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 12 / 35 The 1/3-2/3 Conjecture for finite Coxeter groups Remarkly, the natural generalization of the 1/3-2/3 Conjecture still seems to hold for finite Coxeter group. Conjecture (Gaetz and G. 2020) Let C be a convex set in a finite Coxeter group with jCj > 1, then b(C) ≥ 1=3. There are new equality cases in every type not coming from type A! Convex sets in finite Coxeter groups are exactly convex sets of regions in the corresponding Coxeter arrangement. The conjecture says that there is a hyperplane that cuts the convex set into roughly equal parts. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 13 / 35 Fully commutative elements in Weyl groups Definition (Stembridge 1996) An element w 2 W is fully commutative if no reduced word of w contains a consecutive substring si sj si ··· with mij > 2. | {z } mij m Recall that we have relations (si sj ) ij = id. Example In S4, s2s1s3s2 = 3412 is fully commutative but s2s1s2s3 = 3241 is not. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 14 / 35 Fully commutative elements and width-2 poset Proposition A permutation w 2 Sn is fully commutative if and only if w avoids 321. Definition A poset has width k if its longest antichain has size k. Proposition Under the bijection between (naturally labeled) posets and convex sets in Sn, width-2 posets P correspond to convex sets CP = [id; w]L where w is fully commutative. The 1/3-2/3 Conjecture is known for width-2 poset. We strengthen this result to other Coxeter group. Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 15 / 35 Fully commutative elements in Coxeter group Theorem (Gaetz and G. 2020) Let w 6= id 2 W be a fully commutative element in any Coxeter group W whose Coxeter diagram is acyclic, then b([id; w]L) ≥ 1=3. In particular, this theorem applies to all finite Coxeter groups. Example In the affine Weyl group Af3, w = s1s3s0s2 is fully commutative, but b([id; w]L) = 2=7. s1 • •s2 s0 • •s3 Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 16 / 35 Heaps of fully commutative elements The proof of this theorem uses the theory of heaps. Definition (Stembridge 1996) Let s = si1 ··· si` be a reduced word of w. The Heap poset Hs is the partial order ([`]; ), which is the transitive closure of 2 j k if j ≤ k and (sij sik ) 6= id: In the case where w is fully commutative, the Heap poset does not depend on the choice of a reduced word. We write the Heap poset as Hw . Proposition (Stembridge 1996) Let w 2 W be fully commutative. Then [id; w]L ' J(Hw ): Here J(Hw ) is the lattice of order ideals of Hw . Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 17 / 35 Balance constants for order ideals Let Q be a poset and J(Q) be the order ideals of Q. Define jfJ 2 J(Q) j x 2 Jgj δideal(x; Q) = ; jJ(Q)j b(Q) = max minfδideal(x; Q); 1 − δideal(x; Q)g: x2Q For fully commutative w, we have b([id; w]L) = b(Hw ). Example: non-example Consider the following poset Q = Hw , w = s1s3s0s2 2 W (Af3). We see that jJ(Q)j = 7 and b(Q) = 2=7. 0• •2 1• •3 Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 18 / 35 Equality cases These intervals C = [e; w]L give equality cases b(C) = 1=3: type diagram w Hw • • • A s s s s 2 1 2 1 2 • • •s3 D4 • • s4s2s3s1 • s1 s2 •s 4 • • • • • • B3 s1 s2 s3 s3s2s3s1 • • • •s6 • • E6 • • • • • s6s3s2s4s1s3s5 • • s1 s2 s3 s4 s5 • • • Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 19 / 35 Semiorders Definition A poset P is a semiorder (unit-interval order) if there exists a function f : P ! R such that p < q if and only if f (q) − f (p) > 1.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • Counting Linear Extensions of Sparse Posets⇤
    Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16) Counting Linear Extensions of Sparse Posets⇤ Kustaa Kangas Teemu Hankala Teppo Niinimaki¨ Mikko Koivisto University of Helsinki, Department of Computer Science, Helsinki Institute for Information Technology HIIT, Finland jwkangas,tjhankal,tzniinim,mkhkoivi @cs.helsinki.fi { } Abstract sparse, which raises the question if sparsity can be exploited for counting linear extensions faster. We present two algorithms for computing the num- In this work we present two approaches to counting linear ber of linear extensions of a given n-element poset. n extensions that target sparse posets in particular. In Section 2 Our first approach builds upon an O(2 n)-time we augment the dynamic programming algorithm by splitting dynamic programming algorithm by splitting sub- each upset into connected components and then computing problems into connected components and recursing the number of linear extensions by recursing on each com- on them independently. The recursion may run over ponent independently. We also survey previously proposed two alternative subproblem spaces, and we provide recursive techniques for comparison. heuristics for choosing the more efficient one. Our In Section 3 we show that the problem is solvable in time second algorithm is based on variable elimination t+4 t+4 O(n ), where t is the treewidth of the cover graph. While via inclusion–exclusion and runs in time O(n ), our result stems from a well-known method of nonserial dy- where t is the treewidth of the cover graph. We namic programming [Bertele` and Brioschi, 1972], known demonstrate experimentally that these new algo- as variable elimination and by other names [Dechter, 1999; rithms outperform previously suggested ones for a Koller and Friedman, 2009, Chap.
    [Show full text]
  • Counting Linear Extensions of a Partial Order
    Counting Linear Extensions of a Partial Order Seth Harris March 16, 2011 1 Introduction A partially ordered set (P; <) is a set P together with an irreflexive, transitive relation. A linear extension of (P; <) is a relation (P; ≺) such that (1) for all a; b 2 P either a ≺ b or a = b or b ≺ a, and (2) if a < b then a ≺ b; in other words, a total order that preserves the original partial order. We define Λ(P ) as the set of all linear extensions of P , and define N(P ) = jΛ(P )j. Throughout this paper, we will only be considering finite partial orders on n vertices. The problem of counting linear extensions is not only important in its home field of order theory, but is intimately connected to the theory of sorting. After all, if one is performing a comparison sort, one often starts with incomplete data (a partially sorted set) and must decide what alternatives exist for a total sorting, and whether there is an optimal pair of elements to compare next. Clearly one needs to know the number of possible extensions in order to proceed. It is also common to have partial data but to be unable to compare every pair of data points, in which case one must estimate a total ranking of the set, for instance by ranking the \average height" of an element, its mean rank over all linear extensions. This has widespread applications in the social sciences, such as in ranking lists of consumer products or athletic competitors. But again, to have any hope of calculating this, one must know the (approximate) number of extensions.
    [Show full text]
  • GEOMETRIC EXTENSIONS and the 1/3 — 2/3 CONJECTURE As a Thesis Presented to the Faculty of San Francisco State University P
    GEOMETRIC EXTENSIONS AND THE 1/3 — 2/3 CONJECTURE A s A thesis presented to the faculty of San Francisco State University P I g In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Sam Sehayek San Francisco, California May 2018 Copyright by Sam Sehayek CERTIFICATION OF APPROVAL I certify that I have read GEOMETRIC EXTENSIONS AND THE 1/3 — 2/S CONJECTURE by Sam Sehayek and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfill­ ment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Joseph Gubeladze Professor of Mathematics Matthias Beck Professor of Mathematics Federico Ardila Professor of Mathematics GEOMETRIC EXTENSIONS AND THE 1/3 — 2/3 CONJECTURE Sam Sehayek San Francisco State University 2018 The 1/3—2/3 Conjecture is a famous open problem that deals with partially ordered sets, called posets. Understanding the linear extensions of a poset can unlock hidden structure with interesting applications and further questions. The conjecture says that in every finite poset that is not totally ordered there is a pair x and y, such that x < y in 1/3 to 2/3 of all the linear extensions. Such pairs are called balanced pairs. We develop a geometric version of the conjecture, amenable to computational analysis by considering one dimensions projections of Euclidean realizations of the poset. We confirm the geometric 1/3— 2/3 conjecture for certain classes of posets, for which the original 1/3— 2/3 conjecture is currently out of reach.
    [Show full text]
  • Lipics-Aofa-2018-14.Pdf (0.5
    Beyond Series-Parallel Concurrent Systems: The Case of Arch Processes Olivier Bodini Laboratoire d’Informatique de Paris-Nord, CNRS UMR 7030 - Institut Galilée - Université Paris-Nord, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France. [email protected] Matthieu Dien Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan. [email protected] Antoine Genitrini Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6 -LIP6- UMR 7606, F-75005 Paris, France. [email protected] Alfredo Viola Universidad de la República, Montevideo, Uruguay. viola@fing.edu.uy Abstract In this paper we focus on concurrent processes built on synchronization by means of futures. This concept is an abstraction for processes based on a main execution thread but allowing to delay some computations. The structure of a general concurrent process is a directed acyclic graph (DAG). Since the quantitative study of increasingly labeled DAG (directly related to processes) seems out of reach (this is a ]P-complete problem), we restrict ourselves to the study of arch processes, a simplistic model of processes with futures. They are based on two parameters related to their sizes and their numbers of arches. The increasingly labeled structures seems not to be specifiable in the classical sense of Analytic Combinatorics, but we manage to derive a recurrence equation for the enumeration. For this model we first exhibit an exact and an asymptotic formula for the number of runs of a given process. The second main contribution is composed of a uniform random sampler algorithm and an unranking one that allow efficient generation and exhaustive enumeration of the runs of a given arch process.
    [Show full text]
  • Antimatroids and Balanced Pairs
    Antimatroids and Balanced Pairs David Eppstein Computer Science Department, University of California, Irvine Abstract 1 2 We generalize the 3 – 3 conjecture from partially ordered sets to antimatroids: we conjecture that 1 2 any antimatroid has a pair of elements x;y such that x has probability between 3 and 3 of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antima- troids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements. 1 Introduction The linear extensions of a finite partial order model the sequences that may be formed by adding one element at a time to the end of the sequence, with the constraint that an element may only be added once all its predecessors have already been added. This constraint is monotonic: once an element becomes available to be added, it remains available until it actually is added. However, not every system of monotonic constraints may be modeled by a partial order. Sets of orderings formed by adding one element at a time to the end of a partial sequence, with monotonic constraints that may depend arbitrarily on the previously added elements of the sequence, may instead be modeled in full generality by mathematical structures known as antimatroids. Because antimatroids generalize finite partial orders, they can model the ordering of objects in applica- tions for which partial orders are inadequate, including single-processor scheduling [2], event ordering in discrete event simulations [13], and the sequencing of lessons in automated learning environments [8, 11].
    [Show full text]
  • Permutation Statistics and Linear Extensions of Posets
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series A 58, 85-l 14 (1991) Permutation Statistics and Linear Extensions of Posets ANDERS B3iiRNER Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden AND MICHELLE L. WACHS* Department of Mathematics, University of Miami, Coral Gables, Florida 33124 Communicated by the Managing Editors Received October 4, 1988 We consider subsets of the symmetric group for which the inversion index and major index are equally distributed. Our results extend and unify results of MacMahon, Foata, and Schtitzenberger, and the authors. The sets of permutations under study here arise as linear extensions of labeled posets, and more generally as order closed subsets of a partial ordering of the symmetric group called the weak order. For naturally labeled posets, we completely characterize, as postorder labeled forests, those posets whose linear extension set is equidistributed. A bijec- tion of Foata which takes major index to inversion index plays a fundamental role in our study of equidistributed classes of permutations. We also explore, here, classes of permutations which are invariant under Foata’s bijection. 0 i991 Academic Press, Inc. 1. INTRODUCTION We are primarily concerned here with two permutation statistics of fundamental interest in combinatorics, the inversion index and the major index. We shall view permutations in the symmetric group Yn as words with n distinct letters 1, 2, .. n. More generally, for a set A of n positive integers, yk denotes the set of permutations of A or words with n distinct letters in A.
    [Show full text]
  • Some Aspects of Topological Sorting
    SOME ASPECTS OF TOPOLOGICAL SORTING 1Yohanna Tella (M.Sc, Ph.D), 2 D. Singh (M.Sc, Ph.D) & 3J.N. Singh (M.Sc, Ph.D) 1 (Corresponding Author) Department of Mathematical Sciences Kaduna State University, Kaduna, Nigeria. Email: yot2002ng @kasu.edu.ng 2 (Former Professor, Indian Institute of Technology Bombay). Department of Mathematics Ahmadu Bello University, Zaria, Nigeria. Email: mathdss @yahoo.com 3Department of Mathematics and Computer Science, Barry University, 11300 North East Second Avenue Miami Shores, FL 33161, USA Abstract: In this paper, we provide an outline of most of the known techniques and principal results pertaining to computing and counting topological sorts, realizers and dimension of a finite partially ordered set, and identify some new directions. Key words: Partial Ordering, topological sorting, realizer, dimension. 1. Introduction and principal results pertaining to Topological sorting has been found computing and counting topological particularly useful in sorting and sorts, realizers and dimension of a scheduling problems such as PERT partially ordered set (Poset), and charts used to determine an ordering identify some new directions. of tasks, graphics to render objects 2. Definitions of some terms and from back to front to obscure hidden related results pertaining to surfaces, painting when applying partially ordered sets paints on a surface with various We borrow these definitions from parts and identifying errors in DNA various sources (Brualdi et al., fragment assembly (Knuth, 1973; 1992; Jung, 1992; Trotter 1991). Skiena, 1997; Rosen, 1999). In this Let denote a finite partially paper, we provide systematically an ordered set along with an implicit outline of most of the techniques assumption that denotes the 28 Covenant Journal of Informatics and Communication Technology (CJICT) Vol.
    [Show full text]
  • An Efficient Data Structure for Counting All Linear Extensions of a Poset
    An efficient data structure for counting all linear extensions of a poset, calculating its jump number, and the likes Marcel Wild October 15, 2018 ABSTRACT: Achieving the goals in the title (and others) relies on a cardinality-wise scanning of the ideals of the poset. Specifically, the relevant numbers attached to the k + 1 elment ideals are inferred from the corresponding numbers of the k-element (order) ideals. Crucial in all of this is a compressed representation (using wildcards) of the ideal lattice. The whole scheme invites distributed computation. 1 Introduction The uses of (exactly) counting all linear extensions of a poset are well documented, see e.g. [1] and [9, Part II]. Hence we won't dwell on this, nor on the many applications of the other tasks tackled in this article. Since all of them are either NP-hard or #P-hard [1], we may be forgiven for not stating a formal Theorem; the numerical evidence of efficiency must do (Section 8). It is well known [5, Prop. 3.5.2] that a linear extension corresponds to a path from the bottom to the top of the ideal lattice Id(P ) of P . Attempts to exploit Id(P ) for calculating the number e(P ) of linear extensions of P were independently made in [6], [8], [9]. In the author's opinion they suffer from a one-by-one generation of Id(P ) and (consequently) an inability or unawareness to retrieve all k-element ideals fast. arXiv:1704.07708v1 [cs.DS] 25 Apr 2017 Section 2 offers as a cure the compressed representation of Id(P ) introduced1 in [11].
    [Show full text]