Markov Chains on Graded Posets

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young’s lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.     

Adjoining batch Markov arrival processes of a Markov chain

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D _k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.     

Unit Interval Orders of Open and Closed Intervals

A poset P = (X, ?) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ? y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.     

On the 1/3–2/3 Conjecture

Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x₁x₂?x_n in one-line notation. For distinct elements x, y ∈ P, we define ?(x ? y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\), we say (x, y) is an α-balanced pair if α ≤ ?(x ? y) ≤?1 ? α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.     

Chain Blockers and Convoluted Catalan Numbers

We give new interpretations of Catalan and convoluted Catalan numbers in terms of trees and chain blockers. For a poset P we say that a subset A ? P is a chain blocker if it is an inclusionwise minimal subset of P that contains at least one element from every maximal chain. In particular, we study the set of chain blockers for the class of posets P = C _a × C _b where C _i is the chain 1 < ? < i. We show that subclasses of these chain blockers are counted by Catalan and convoluted Catalan numbers.     

Posets with Cover Graph of Pathwidth two have Bounded Dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S ₅ as a subposet, then the cover graph of P has treewidth at least 3.     

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset P, denoted \(\dim (P)\), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with \(|P|\le 2n+1\) is n, provided \(n\ge 2\), and this inequality is tight when P contains the standard example \(S_n\). However, there are posets with large dimension that do not contain the standard example \(S_2\). Moreover, for each fixed \(d\ge 2\), if P is a poset with \(|P|\le 2n+1\) and P does not contain the standard example \(S_d\), then \(\dim (P)=o(n)\). Also, for large n, there is a poset P with \(|P|=2n\) and \(\dim (P)\ge (1-o(1))n\) such that the largest d so that P contains the standard example \(S_d\) is o(n). In this paper, we will show that for every integer \(c\ge 1\), there is an integer \(f(c)=O(c^2)\) so that for large enough n, if P is a poset with \(|P|\le 2n+1\) and \(\dim (P)\ge n-c\), then P contains a standard example \(S_d\) with \(d\ge n-f(c)\). From below, we show that \(f(c)={\varOmega }(c^{4/3})\). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.     

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ) and the stopping probabilities P(τ = τA)with A ∈ C, where τA is the waiting time until the pattern A appears as a run.     

Boolean Lattices: Ramsey Properties and Embeddings

A subposet Q ^′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q ^′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P ^′, let the poset Ramsey number R(P, P ^′) be the smallest N such that no matter how the elements of the Boolean lattice Q _N are colored red and blue, there is a copy of P with all red elements or a copy of P ^′ with all blue elements. We provide some general bounds on R(P, P ^′) and focus on the situation when P and P ^′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Q _n in Q _N and for a multicolor version of a poset Ramsey number.     

Planar Posets,Dimension, Breadth and the Number of Minimal Elements

In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t + 3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f(d) so that if P is a planar poset with dim(P) ≥ f(d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f, if it exists, satisfies f(d) ≥ d + 2. Here, we show that lim _d→∞ f(d)/d ≥ 2.     

Peckness of Edge Posets

For any graded poset P, we define a new graded poset, ??(P), whose elements are the edges in the Hasse diagram of P. For any group G acting on the boolean algebra B _n in a rank-preserving fashion we conjecture that ??(B _n /G) is Peck. We prove that the conjecture holds for “common cover transitive” actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.     

A generalization of the Dedekind–MacNeille completion

In this paper we introduce the notion of \(Z_{\delta }\)-continuity as a generalization of precontinuity, complete continuity and \(s_{2}\)-continuity, where Z is a subset selection. And for each poset P, a closure space \(Z^{c}_{\delta }(P)\) arises naturally. For any subset system Z, we define a new type of completion, called \(Z_{\delta }\)-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then \(cl_{Z}(\Psi (P))\), the Z-closure of all principal ideals \(\Psi (P)\) of poset P in \(Z^{c}_{\delta }(P)\), is a \(Z_{\delta }\)-completion of P and \(Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))\); (2) let Z be an HUL-system and P a \(Z_{\delta }\)-continuous poset, then the \(Z_{\delta }\)-completion of P is also \(Z_{\delta }\)-continuous, and a Z-complete poset L is a \(Z_{\delta }\)-completion of P iff P is an embedded \(Z_{\delta }\)-basis of L; (3) the Dedekind–MacNeille completion is a special case of the \(Z_{\delta }\)-completion.     

Dimension and Matchings in Comparability and Incomparability Graphs

We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d=3, then there is a matching of size d in the comparability graph of P. There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for which the maximum matching in the cover graph of P has size \(O(\log d)\). On the other hand, there is a dual result in which the role of chains and antichains is reversed, as we show that there is also a matching of size d in the incomparability graph of P. The proof of the result for comparability graphs has elements in common with Perles’ proof of Dilworth’s theorem. Either result has the following theorem of Hiraguchi as an immediate corollary: \(\dim (P)\le |P|/2\) when |P|=4.     

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern p is denoted A v _P (p). We extend a proof of Simion and Schmidt to show that A v _P (132)=A v _P (123) for any poset P, and we exactly classify the posets for which equality holds.     

On structural decompositions of finite frames

A frame in an n-dimensional Hilbert space H _n is a possibly redundant collection of vectors {f _i } _i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f _i } _i∈I is said to be scalable if there exist nonnegative scalars {c _i } _i∈I such that {c _i f _i } _i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f _i } _i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J?I is in the factor poset iff {f _j } _j∈J is a tight frame for H _n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability.     

Poset Entropy Versus Number of Linear Extensions: The Width-2 Case

Kahn and Kim (J. Comput. Sci. 51, 3, 390–399, 1995) have shown that for a finite poset P, the entropy of the incomparability graph of P (normalized by multiplying by the order of P) and the base-2 logarithm of the number of linear extensions of P are within constant factors from each other. The tight constant for the upper bound was recently shown to be 2 by Cardinal et al. (Combinatorica 33, 655–697, 2013). Here, we refine this last result in case P has width 2: we show that the constant can be replaced by 2?ε if one also takes into account the number of connected components of size 2 in the incomparability graph of P. Our result leads to a better upper bound for the number of comparisons in algorithms for the problem of sorting under partial information.     

The Factorization of Queueing Equations and Their Interpretation

There are many queueing systems, including the M ^x /M ^y /c queue, the GI ^x /M/c queue and the M/D/c queue, in which the distribution of the queue length at certain epochs is determined by a Markov chain with the following structure. Except for a number of boundary states, all columns of the transition matrix are identical except for a shift which assures that there is always the same element occupying the main diagonal. This paper describes how one can find the equilibrium distribution for such Markov chains. Typically, this problem is solved by factorizing of a certain expression characterizing the repeated columns. In this paper, we show that this factorization has a probabilistic significance and we use this result to develop new approaches for finding the equilibrium distribution in question.     

On the Spectrum of the Generalised Petersen Graphs

We show that the gap between the two greatest eigenvalues of the generalised Petersen graphs P(n, k) tends to zero as \(n \rightarrow \infty \). Moreover, we provide explicit upper bounds on the size of this gap. It follows that these graphs have poor expansion properties for large values of n. We also show that there is a positive proportion of the eigenvalues of P(n, k) tending to three.     

Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound \({\mathrm La}(n,P)\le \frac {1}{2}(|P|+h-2)\left (\begin {array}{c}n \\ \lfloor {n/2}\rfloor \end {array}\right )\) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper bound

$${\mathrm La}(n,P)\le \left( \frac{|P|+h-\alpha(G_{P})-2}{2}\right)\left( \begin{array}{c}n \\ \lfloor{\frac{n}{2}}\rfloor\end{array}\right) $$

for any graded poset P, where α(G _P ) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

     

A Markov copula model with regime switching and its application

Regime switching, which is described by a Markov chain, is introduced in a Markov copula model. We prove that the marginals (X,H ⁱ ), i = 1, 2, 3 of the Markov copula model (X,H) are still Markov processes and have martingale property. In this proposed model, a pricing formula of credit default swap (CDS) with bilateral counterparty risk is derived.