A recurrence for linear extensions

The number e(P) of linear extensions of a finite poset P is expressed in terms of e(Q) for certain smaller posets Q. The proof is based on M. Schützengerger's concept of promotions of linear extensions.Partially supported by NSF Grant #DMS-8700995.Partially supported by NSF Grant #DMS-8401376.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

Representing an ordered set as the intersection of super greedy linear extensions

A linear extension [x ₁2<...t] of a finite ordered set P=(P, <) is super greedy if it can be obtained using the following procedure: Choose x ₁ to be a minimal element of P; suppose x ₁,...,x _i have been chosen; define p(x) to be the largest ji such that x _jj exists and 0 otherwise; choose x _i+1 to be a minimal element of P-{ x ₁,...,x _i} which maximizes p. Every finite ordered set P can be represented as the intersection of a family of super greedy linear extensions, called a super greedy realizer of P. The super greedy dimension of P is the minimum cardinality of a super greedy realizer of P. Best possible upper bounds for the super greedy dimension of P are derived in terms of |P-A| and width (P-A), where A is a maximal antichain.Research supported in part by NSF grant IPS-80110451.Research supported in part by ONR grant N00014-85K-0494 and NSERC grants 69-3378, 69-0259, and 69-1325.Research supported in part by NSF grant DMS-8401281.     

Greedy linear extensions for minimizing bumps

Let P be a finite poset and let L={x ₁<...n} be a linear extension of P. A bump in L is an ordered pair (x _i , x _i+1) where x _ii+1 in P. The bump number of P is the least integer b(P), such that there exists a linear extension of P with b(P) bumps. We call L optimal if the number of bumps of L is b(P). We call L greedy if x _{i j} for every j>i, whenever (x _i, x _i+1) is a bump. A poset P is called greedy if every greedy linear extension of P is optimal. Our main result is that in a greedy poset every optimal linear extension is greedy. As a consequence, we prove that every greedy poset of bump number k is the linear sum of k+1 greedy posets, each of bump number zero.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

Substitution and atomic extension on greedy posets

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, if P=(P ₁ , x, P ₂ )is a greedy poset, then P ₁ and P ₂ are greedy posets, the converse being false. However, for the atomic extension, P=P ₁ (x, P ₂ )is a greedy poset if and only if P ₁ and P ₂ are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containing M _n (n2), B ₃ and closed by atomic extension. The class C _n of greedy posets with jump number n is infinite. However, we show that C _n can be obtained, in a very simple way, from a subclass D _n of finite cardinal ity. We construct D _n for n=1, 2.     

Greedy posets for the bump-minimizing problem

A bump (x _i,x _i+1) occurs in a linear extension L={x ₁<...n} of a poset P, if x _ii+1 in P. L. is greedy if x _ij for every j>i, whenever (x _i x _i+1) in a bump in L. The purpose of this paper is to give a characterization of all greedy posets. These are the posets for which every greedy linear extension has a minimum number of bumps.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

Balance theorems for height-2 posets

We prove that every height-2 finite poset with three or more points has an incomparable pair {x, y} such that the proportion of all linear extensions of the poset in which x is less than y is between 1/3 and 2/3. A related result of Komlós says that the containment interval [1/3, 2/3] shrinks to [1/2, 1/2] in the limit as the width of height-2 posets becomes large. We conjecture that a poset denoted by V _m ⁺ maximizes the containment interval for height-2 posets of width m+1.     

3-Interval irreducible partially ordered sets

In this paper we discuss the characterization problem for posets of interval dimension at most 2. We compile the minimal list of forbidden posets for interval dimension 2. Members of this list are called 3-interval irreducible posets. The problem is related to a series of characterization problems which have been solved earlier. These are: The characterization of planar lattices, due to Kelly and Rival [5], the characterization of posets of dimension at most 2 (3-irreducible posets) which has been obtained independently by Trotter and Moore [8] and by Kelly [4] and the characterization of bipartite 3-interval irreducible posets due to Trotter [9].We show that every 3-interval irreducible poset is a reduced partial stack of some bipartite 3-interval irreducible poset. Moreover, we succeed in classifying the 3-interval irreducible partial stacks of most of the bipartite 3-interval irreducible posets. Our arguments depend on a transformationP B(P), such that IdimP=dimB(P). This transformation has been introduced in [2].Supported by the DFG under grant FE 340/2–1.     

Linear extension majority cycles in height-1 orders

For points x and y in a poset (X, >) let x> _p y mean that more linear extensions of the poset have x above y than y above x. It has been known for some time that > _p can have cycles when the height of the poset is two or more. Moreover, the smallest posets with a > _p cycle have nine points and heights of 2, 3 and 4. We show here that height-1 posets can also have > _p cycles. Our smallest example for this phenomenon has 15 points.Research supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Dimension of the crown Skn

In 1941, Dushnik and Miller introduced the concept of the dimension of a poset (X, P) as the minimum number of linear extensions of P whose intersection is exactly P. Although Dilworth has given a formula for the dimension of distributive lattices, the general problem of determining the dimension of a poset is quite difficult. An equally difficult problem is to classify those posets which are dimension irreducible, i.e., those posets for which the removal of any point lowers the dimension. In this paper, we construct for each n≥3, k≥0, a poset, called a crown and denoted S^k_n, for which the dimension is given by the formula $\text{2?(n+k)}\text{(k+2)}$. Furthermore, for each t≥3, we show that there are infinitely many crowns which are irreducible and have dimension t. We then demonstrate a method of combining a collection of irreducible crowns to form an irreducible poset whose dimension is the sum of the crowns in the collection. Finally, we construct some infinite crowns possessing combinatorial properties similar to finite crowns.     

A structure theorem for posets admitting a “strong” chain partition:: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

Suppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q∈P lie in distinct chains and p<q, then every other element of P is either above p or below q.In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets such that, for 1?i?n, one of the following occurs:

(1)

R_i is disjoint from one of the chains of the partition; or

(2)

if p, q∈R_i are in distinct chains, then they are incomparable.

The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets.In 1996, a proof of the Daykin-Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified.In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained.     

Posets with large dimension and relatively few critical pairs

The dimension of a poset (partially ordered set)P=(X, P) is the minimum number of linear extensions ofP whose intersection isP. It is also the minimum number of extensions ofP needed to reverse all critical pairs. Since any critical pair is reversed by some extension, the dimensiont never exceeds the number of critical pairsm. This paper analyzes the relationship betweent andm, when 3tmt+2, in terms of induced subposet containment. Ifmt+1 then the poset must containS _t , the standard example of at-dimensional poset. The analysis form=t+2 leads to dimension products and David Kelly's concept of a split. Whent=3 andm=5, the poset must contain eitherS ₃, or the 6-point poset called a chevron, or the chevron's dual. Whent4 andm=t+2, the poset must containS _t , or the dimension product of the Kelly split of a chevron andS _t–3, or the dual of this product.     

Angle orders

A finite poset is an angle order if its points can be mapped into angular regions in the plane so thatx precedesy in the poset precisely when the region forx is properly included in the region fory. We show that all posets of dimension four or less are angle orders, all interval orders are angle orders, and that some angle orders must have an angular region less than 180° (or more than 180°). The latter result is used to prove that there are posets that are not angle orders.The smallest verified poset that is not an angle order has 198 points. We suspect that the minimum is around 30 points. Other open problems are noted, including whether there are dimension-5 posets that are not angle orders.Research supported in part by the National Science Foundation, grant number DMS-8401281.     

A linear time algorithm to find the jump number of 2-dimensional bipartite partial orders

Let L=u ₁ , u ₂ , ..., u _k be a linear extension of a poset P. Each pair (u _i , u _i+1 ) of unrelated elements in P is called a jump of L. The jump number problem is to find L with the minimum number of jumps. The problem is known to be NP-hard even on bipartite posets. Here we present a linear time algorithm for it in 2-dimensional bipartite posets. We also discuss briefly some weighted cases.     

On the Boolean dimension of spherical orders

It is well known that if a planar order P is bounded, i.e. has only one minimum and one maximum, then the dimension of P (LD(P)) is at most 2, and if we remove the restriction that P has only one maximum then LD(P)3. However, the dimension of a bounded order drawn on the sphere can be arbitrarily large.The Boolean dimension BD(P) of a poset P is the minimum number of linear orders such that the order relation of P can be written as some Boolean combination of the linear orders. We show that the Boolean dimension of bounded spherical orders is never greater than 4, and is not greater than 5 in the case the poset has more than one maximal element, but only one minimum. These results are obtained by a characterization of spherical orders in terms of containment between circular arcs.Part of this work was carried out while both authors were visiting the Department of Applied Mathematics (KAM) of Charles University, Prague. The authors acknowledge support from the EU HCM project DONET.     

Jump number problem: The role of matroids

Let L=x ₁ x ₂...x _n be a linear extension of a poset P. Each pair (x _i , x _i+1) such that x _i x _i+1in P is called a jump of L. It is well known that for N-free posets a natural greedy procedure constructing linear extensions yields a linear extension with a minimum number of jumps. We show that there is a matroid corresponding to any N-free poset and apply the Rado-Edmonds Theorem to obtain another proof of this result.     

Inequalities for the greedy dimensions of ordered sets

Every linear extension L: [x ₁<x ₂<...<x _m ] of an ordered set P on m points arises from the simple algorithm: For each i with 0i<m, choose x _i+1 as a minimal element of P–{x _j :ji}. A linear extension is said to be greedy, if we also require that x _i+1 covers x _i in P whenever possible. The greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. In this paper, we develop several inequalities bounding the greedy dimension of P as a function of other parameters of P. We show that the greedy dimension of P does not exceed the width of P. If A is an antichain in P and |P–A|2, we show that the greedy dimension of P does not exceed |P–A|. As a consequence, the greedy dimension of P does not exceed |P|/2 when |P|4. If the width of P–A is n and n2, we show that the greedy dimension of P does not exceed n ²+n. If A is the set of minimal elements of P, then this inequality can be strengthened to 2n–1. If A is the set of maximal elements, then the inequality can be further strengthened to n+1. Examples are presented to show that each of these inequalities is best possible.Research supported in part by the National Science Foundation under ISP-80110451.Research supported in part by the National Science Foundation under ISP-80110451 and DMS-8401281.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order defined onP so thatab holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.This research was written while the first author was visiting the University of Calgary.Research supported by NSERC grant #69-0982.