A Combinatorial Proof for the Rank-Unimodality of Poset Order Ideals

Transcription

1 A Combinatorial Proof for the Rank-Unimodality of Poset Order Ideals By: Kevin Rao Clements High School, Sugar Land, Texas Mentored by Dr. Edward Early of St. Edwards University

2 Special thanks to Hans Li and William Liu for their contributions to this project.

3 A Combinatorial Proof for the Rank-Unimodality of Poset Order Ideals Abstract Posets are sets with ordering relations between some of its elements. This construction is a central topic in combinatorics and lattice theory, and posets also serve as versatile models for the distribution and prioritization of tasks, thus making them important in parallel computing. Our research deals with order ideals of the product of linear posets called chains (denoted n n n 3... ). An order ideal is a set of elements and all elements less than them. The poset of all order ideals of a poset P is denoted L(P ), and is arranged in levels. We aim to show that these levels, or the structure of L(n n n 3... ), are rank unimodal (the size of each level increases to a single local maximum and then decreases). We began our research by improving a standard poset algorithm and demonstrating rank unimodality in novel infinite families of posets. From our data, we gained insight on the structure of higher dimensional posets. Then we found an ingenious bijection between L(n n n 3... ) and W (n n n 3... ), which is the first proof that holds for all possible products of chains. Applying our bijection, we proved our second result that the product of any three chains is rank-unimodal, making the biggest breakthrough since O Hara s 990 proof of two-chain posets. Finally, we extended all of these results to outline a proof for the rank unimodality of infinite dimensional posets.

4 Introduction Posets are sets with relations between its elements that indicate order. They serve as versatile models for the distribution and prioritization of tasks. Applying these mathematical models to synchronize parallel computing, internet search engines such as Google Search are able to produce results at record-breaking speed []. Working hand in hand with parallel computing, poset theory also allows for highly efficient multitasking on handheld devices such as smartphones, where increasing computation speed is not a matter of adding more CPUs. With new software, smartphones can greatly increase their capability to handle many apps running at the same time []. Our proofs and deeper understanding of special poset properties will help unleash the full potential of parallel computing and shed light on exciting new possibilities. Our research is on the unimodality of a type of poset called poset order ideals. Posets are arranged in levels of order, and unimodality is a property such that when moving up the levels, the size of each level increases up to a point, and decreases back down symmetrically. Proving that these posets are unimodal may lead to greater optimization within parallel computations [3]. Previous work on this subject, beginning with Stanley in 970, only touched on simple posets [4]. Since then, some researchers have studied a few infinite families, but their results were limited to one or two dimensions [5]. In this paper, we first summarize poset theory and its relevant definitions as well as analyze important pertinent theorems. Then, from the data generated with the new algorithm we developed, we present several infinite families of posets that we showed to be unimodal. Next, we simplify the problem of unimodality with

5 a bijection to a simpler problem. Using this result, we present the first pure combinatorics proof for the unimodality of three dimensional posets. Finally, we present a promising avenue for future research on the rank unimodality of infinite posets. Definitions. Poset Theory A partially ordered set, or poset, is a set of elements where some elements are comparable with one another, while others are unrelated. This comparison, denoted, follows several rules: if x and y are elements in a poset then x x; if x y and y x then x = y; and if x y and y z, then x z. Two elements x and y are called incomparable if they cannot be ordered with the rule. In other words, x y and y x [4]. Posets can be visually represented by graphs called Hasse diagrams (Figure ). The Hasse diagram of a poset P is a graph whose nodes are elements of P, edges are comparison relations, and y appears above x for all elements x and y such that x < y. In a ranked poset, each node has a well-defined level, or distance from the bottom of a Hasse diagram. The rank of a level is the number of nodes that exist on that level [4] Figure : Poset of divisors of in a Hasse diagram.

6 For example in the Hasse diagram above the set of divisors of is stored in a poset. The edges represent divisibility relations, and there are 4 levels, with ranks,,, respectively. Note that and are comparable although there does not exist a direct connection between the two, due to the path going up through 4 that leads to. However, and 3 are not comparable because there is no path or connection between the nodes. The specific posets we are working with in our research are lattices consisting of the direct product of chains. A chain is simply a totally-ordered poset arranged in a straight line. For chains A and B, their direct product A B is the poset of points {(a, b) a A, b B}, and (a, b ) (a, b ) if and only if a a and b b. Structurally, the direct product of chains is a rectangular grid rotated such that a corner is the bottommost node and the opposite corner is the topmost node. The direct product of more than chains is a higher dimensional grid, but the Hasse diagram can still be represented in dimensions and there still is only a single top and bottom node.. Order Ideals An order ideal is a set of one or more elements such that if a is an element of the order ideal and b a, then b is also an element of the order ideal [6]. In other words, an order ideal is a set of nodes and all nodes ordered below them. In Figure below, the red nodes of the lattice represents an order ideal that satisfies this rule: no red node is above a black node. 3

7 Figure : An order ideal of the 4 7 poset Let L(n, n, n 3,... n k ) be the poset of order ideals for the product of chains n n... n k. The poset of order ideals stores all order ideals for a specific poset, such that an order ideal with k elements is on the k + level and two elements are comparable if one is a subset of another. An example is shown in Figure 3 below. 4 {,,3,4} {,,3} Level 5 Level 4 3 {,} {,3} Level 3 {} {} Level Level Figure 3: lattice and its corresponding poset order ideals, arranged in a lattice In Figure 3, the Hasse diagram on the right represents the poset of order ideals for a lattice. Each level in the poset contains all order ideals of cardinality one less than the number of the level. Order ideals have many interesting properties and applications, 4

8 and proving features about it is a central topic to poset theory. One conjectured property is unimodality. A sequence a, a,... a k is defined to be unimodal if there exists an element a i such that a a a 3 a i a i+ a i+ a k a k. [6] In other words, unimodality means that there exists a single local maximum. A poset is defined to be rank-unimodal if the ranks of the levels going from top to bottom are unimodal and rank-symmetric if the ranks are symmetric. Looking at the previous diagram, we see that the ranks of the L(, ) poset are,,,, which is unimodal. Our project s aim is to show that L(P ) is rank-unimodal for all posets P that are products of chains..3 Linear Extensions and Descents Suppose there are n nodes in a lattice. These nodes can then be labelled to n in a number of ways, with the only rule being that every node s label must be greater than that of the nodes it covers. Such a labelling is called a linear extension [6]. Below in Figure 3 are examples of legal and illegal numberings. 5

9 6 Legal Illegal Figure 4: Legal and illegal numberings, where illegal edges are circled in red A descent is defined to be a location where a larger number precedes a smaller number in a string of numbers. For example, the sequence,, 5, 4, 3, 6 has a descent in position 3 because 5 is greater than 4 and appears earlier in the sequence. There is another descent in position 4 by similar reasoning. Suppose we have a poset P with n elements. From each linear extension of P, we can obtain a permutation of the numbers to n called a descent chain. This is done by writing the labels of the nodes in the default labelling in the order of the nodes in the linear extension. Each descent chain describes a distinct linear extension. 6

10 Default Labelling Linear Extension Figure 5: Example linear extension of a 3 lattice For example, we consider a linear extension of a 3 lattice in Figure 5. The labelling on the left is default, and the labelling on the right is the linear extension. To obtain the descent chain from the linear extension, we read off the value of each node of the default labelling in the order that the linear extension specifies. In this case, the descent chain is, 3, 5,, 4, 6 because of the position of the numbers,, 3, 4, 5, 6 in the linear extension..4 W Polynomials We use what is called a W polynomial as a bookkeeping tool for the location and number of descents. W polynomials are sums of powers of an arbitrary variable q. Each descent chain generates a term, with the power of the term representing the sum of the locations of the descents for the descent chain. For example, a descent chain with descents at position 4 and 6 would correspond to the term q 0. W s (P ) is the sum of all such terms from descent chains with s descents for a given poset P. 7

11 Rank generating functions are polynomials in which the coefficient of each term represents the rank of each level on a poset. For example, the fank generating function +q+q +q 3 +q 4 would correspond to a poset with one element on the bottom row, one on the second row, two on the third row and so on. W polynomials are important because they have been shown to be closely related to order ideals, and actually serve as the basis for our proof of the unimodality of order ideals. By analyzing these generating functions, we can determine whether a poset of order ideals is unimodal or not. 3 Previous Work In the past, the unimodality of L has only been proven for the product of chains. Kathy O Hara proved the unimodality of L(m, n) using combinatorical methods [5], while Edward Early proved L(, m, n) was unimodal as well with a combinatorical proof [6]. So far, no combinatorical proof exists for a product of 3 or more general chains. We aim to prove a new result: that L is unimodal for the product of 3 chains. 3. Stanley s Theorem Let P be a poset of size p consisting of the product of k chains, and Q be a poset equal to P multiplied by another chain of arbitrary length m. Then, Stanley s theorem states: p L(Q) = p + m s W s (P ) s=0 p This theorem is significant because it shows a relationship between order ideals and W q 8

12 polynomials: If the W polynomial of the product of k chains is unimodal, then the poset of order ideals of the same k chains times a chain of any length is also unimodal. Thus, if there was a way to prove W polynomials were unimodal, then it would also prove infinite families of poset order ideals were also unimodal. 4 Current Progress In order to gain insight into this problem, we improved the standard algorithm and developed software tools to generate large amounts of data, demonstrating rank unimodality in novel infinite families of posets. Inspired by the results of the programs, we proved that W (P ) = L(P ) for all P equal to a product of any amount of chains. The generality of this result is significant, as before all theorems were limited to a product of only or 3 chains. By using this result, we then found the first combinatorial proof of the unimodality of the lattice of order ideals for a product of 3 chains. 4. Computer Analysis From the poset package created in Maple by John Stembridge, we developed an improved algorithm to better understand the structure of these posets []. Stanley s Theorem states that if W s is known to be unimodal for a product of chains, then L is unimodal for the product of those specific chains and a general chain. This means that we can simply use the computer algorithm to show W s was unimodal to find the following infinite families of unimodal order ideals: L(,, k, m), for k 0 9

13 L(3, 3, 3, m), L(3, 3, 4, m) L(,,, m, n) for m 4 After analyzing pages of data, we noticed many recurring patterns and relations. For example, using our program we found the generating function of W (,, 4) was equal to: q 4 + 3q 3 + 6q + 7q + 0q 0 + 0q 9 + q 8 + 0q 7 + 0q 6 + 7q 5 + 6q 4 + 3q 3 + q and L(,, 4) was equal to: q 6 +q 5 +3q 4 +4q 3 +7q +8q +q 0 +q 9 +3q 8 +q 7 +q 6 +8 q 5 +7q 4 +4q 3 +3q +q+ After computing more cases, we observed that the coefficients of W (P ) were always one less than the coefficients of L(P ) for all posets P. This observation eventually became our Theorem, and led us to devise a combinatorial proof of it. 4. Theorem - W (P ) = L(P ) Theorem states that the rank generating function of W (P ) is equal to the rank generating function of L(P ) with each coefficient reduced by one. To prove this, we draw a bijection between descent chains with one descent and non-consecutive order ideals. First, we show that every order ideal produces a distinct descent chain. Suppose the order ideal has n elements. Take the nodes of the order ideal elements, and arrange them in increasing order. Then, take the remaining values of the nodes and append them to the end in increasing order. For the example in Figure 6, this would give the sequence, 3, 5, 7,, 4, 6, 8. Unless our order ideal is,, 3,, n, this results in exactly one descent because the chain is completely in order except for the position where we append the two sections. We then draw the linear extension corresponding to this descent chain, and show 0

14 that it is indeed legal. The nodes indicated by the first n descent chain digits represent an order ideal - thus, they cover all lower elements. Therefore, in addition to being ordered, they do not leave any gaps for the greater numbers to fill in. That is, the rest of the digits, aside from the first n, correspond to nodes higher than those of the first n. Because these nodes are also numbered in increasing order, they too form a legal labelling. Thus the unique labelling is a legal one. To complete the bijection we must prove every descent chain with one descent produces an order ideal. Consider a linear extension with one descent in its chain. Split it where the descent occurs. This produces two increasing sequences of digits, the first of which forms an order ideal. For the case of Figure 6, it produces the two sequences, 3, 5, 7 and, 4, 6, 8. To see this, note that the first n nodes of the descent chain correspond to the nodes labeled,,..., n on the natural labelling. By the definition of natural labelling, the first n nodes must be an order ideal Figure 6: Renumbering using the {, 3, 5, 7} poset

15 4.3 Theorem - L(k, m, n) is unimodal Let P be a poset of cardinality N consisting of the product of 3 chains of any integer length k, m, n. From Theorem, we know that the rank generating function of W (P ) is simply the rank generating function of L(P ) with a one subtracted from each term. Thus, proving W (P ) is unimodal implies the unimodality of L(P ). The rank generating function of W (P ) stores the number of linear extensions of P with only one descent, such that the s th term of the rank generating function equals the number of linear extensions with a descent at position s +. To prove that W (P ) is unimodal, we must show the number of linear extensions with a descent at position s increases as s increases until s reaches N, at which the number of linear extensions with a descent at position s begins decreasing. There are only 3 types of descents, all of which can be shown to satisfy this property, thus proving unimodality Type Descents - Same Level The first type of descent occurs on the same level in the linear extension. Note that for a linear extension to have a descent at position s, then s + must occur to the left or below s in the labelling. For example, in Figure 6 below, the 7 appears to the left of the 6 in the linear extension on the right. Thus, the descent chain is,, 3, 4, 6, 7, 5, 8, 9, 0,, and has a descent at position 6 (7, 5). That means if s and s + are on the same level, then we can have a descent simply by rearranging the level such that s + is to the left of s and the other N nodes are arranged in order. This means the number of first type descents at position s increases with s because as s increases the number of possible places the s + can

16 exist increases Default Descent at 6 Figure 7: Type descent - s and s + on the same level 4.3. Type Descents - Adjacent Levels The second type of descent occurs when either s or s + are one level away from their original level. These occur when s moves up a level or s + moves down a level, and the rest of the N nodes are labeled in order. Either way, it forces s + to be below s in the linear extension, which creates a descent. For example, to make a descent at position 6 from the default labelling in Figure 6 above, the 6 could either move up one level or the 7 moves down a level. However, there are considerations that must be made. The move up or down must result in a labelling that still follows the ordering rule. Thus, the number of moves are limited by the magnitude of s. If s is too large, then s + cannot be moved down without being so large that the numbers above s + are not greater than s +. If s is too small, then s cannot be moved up without being so small that the numbers below it aren t smaller 3

17 than it. Because P is a product of chains, it is known to be unimodal. Thus, the rank of the levels increases as s increases. This means that as s increases, there are more places for s to swap upwards, and these new openings grow faster than are lost because s + grows too large to move down. Thus, as s increases, the number of linear extensions with type descents at s has a net increase as well Descent at 6 Descent at 6 Descent at 7 Figure 8: Type 3 descent - s and s + are more than level apart Type 3 Descents - Disconnected Levels The last type of descent occurs when s and s + are moved more than one level away from one another. Note that using the previous type of descent does not work because if s + is moved or more levels down, it will have to be greater than every element on the level above it. Similarly if s is moved or more levels up, it will be smaller than every element on the level below it. Thus the only way to have s + multiple levels below s is if all the smaller numbers were grouped together in a closed group and all the larger numbers were ordered above them. In the example shown in Figure 7 on the left, this can be achieved 4

18 by numbering the left face with the smaller numbers through 6, and then numbering the rest of the larger numbers in the remaining available nodes. Similarly, the middle example numbers the right face from through 6 and fills in the remaining nodes. This works because the faces only have to be greater than nodes in the same face, and are closed from the rest of the poset. Furthermore, if s is large enough then the smaller numbers can be used to fill out the lower levels and then proceed up a face, as shown in the example on the right. The bottom levels are filled completely before the rest of the numbers before s are used to number the left face, resulting in more labellings. Thus, as s increases, the number of possible linear extensions with this type of descent increases. Because for 3 types of descents their respective linear extensions increase in number as s increases, this means that W (P ) increases. However, we haven t proven that the second half of W (P ) decreases in number. To do this, we simply show that W (P ) is symmetric, meaning if the first half increases then the second half must decrease. Once that is shown, then the proof is complete Lemma - W s is symmetric around q ns Consider a descent chain of length n with s descents and its corresponding linear extension. k i Such a descent chain with descents at k, k,... k s contributes the term qi= to W s. By looking at the labels of the original labelling in reverse order, we get a string S that is the original descent chain in reverse order. We then subtract each term in S from n + s to get S. Reading the labels of the lattice in reverse is equivalent to rotating the twodimensional representation of the lattice 80 degrees and subtracting each term from n +. This new lattice has a descent chain equal to S. In short, we can see that for every descent 5

19 chain, there exists another descent chain whose descent locations are a mirror image. This second descent chain has descents at locations n k, n k,... n k s and contributes the term q s n k i i= = q ns s k i i=. Using this method, we can see that for every term q k in our W s polynomial, there exists a term q ns k. It follows that the polynomial W s is centered around the term kq ns. 5 Conclusion Before our research, mathematicians working on posets only proved a few minor theorems. With Theorem, we established the very first theorem that is true for all possible posets, removing obsolete limits to using posets and widening the path for future applications. Along the way, we discovered and proved several major properties of posets. To make our research possible, we optimized a standard poset algorithm and developed software tools in order to analyze a wider range of posets. Based on the large amount of data we generated, we discovered new infinite families of posets and demonstrated their rank-unimodality. Our research led us to notable discoveries and solid proofs for posets that lay a firm foundation for future work. 5. Applications Posets are a fundamental concept in mathematics. In many ways, posets are considered a prime unifying structure of combinatorics and other fields [9][0]. For example, in 98, Robert Proctor discovered that proving the unimodality of order ideals would also prove a significant unsolved problem in linear algebra and lattice theory [8]. Without posets, the 6

20 connection between these diverse areas of math would still be lost. In the information age, the development of more powerful monolithic CPU hardware can no longer keep up with the surging need of larger computations. Distributed parallel computing is the key to solving this problem. It splits a big problem into pieces, handles them in multiple computers simultaneously, then stitches the results together. In this way, a number of commodity computers can achieve the same, and often even greater, total power as the greatest supercomputer. Poset order ideals model the process of breaking a large job into smaller tasks as well as fitting the results together. This is done by representing each task as an element in a poset. The poset of order ideals represents every possible way to split the problem up, solve each piece, and put the end results back together. Each traversal of the poset from bottom to top represents a different way to solve the problem, where some tasks must be completed before others. When the poset is traversed from bottom to top, the order ideals get progressively larger and more complex as more and more small tasks are added on. The process continues until the final result, represented by the complete order ideal, is produced. In this process, different schemes of coordinating the tasks can be analyzed and compared to devise an optimal plan for reaching the desired objective [3]. On a large scale, posets may help improve the speed of distributed database searches, such as Google Search []. On the other hand, they are particularly significant for improving the performance of smartphones that run many apps in parallel, where the room to physically expand computing power is rather limited []. Our proofs and deeper insight of posets lay a solid foundation for further researches and open doors to exciting new possibilities. 7

21 5. Future Work In the future, we hope to show that the product of an infinite number of any size chains is rank unimodal. The most promising component of our results is Theorem, which relates the unimodality of W and order ideals. If W (P ) is shown to be unimodal for all P, by strong induction and Stanley s Theorem, we can show that all posets are rank-unimodal. That is, the unimodality a product of one chain L(m) would show that all W n (m) are unimodal, which by Stanley s Theorem means that all products of two chains L(m, n) are unimodal. Again, using the generalized relation, we could show that W n (m n) is unimodal, which implies that the product of three chains L(k, m, n) is unimodal. By continuing this process, we can show that the product of any amount of chains has unimodal order ideals. In using our Maple code, we have found an interesting result, which holds for all verifiable cases: W s ( n) = L(s (n s )) L(s (n s)). This result indicates there are ways of proving q s(s+) L(s) W s is unimodal for all s, which would complete the proof outlined above. Through greater analysis of the nature of order ideals using our program, if given more time, we can find the relation that proves all posets have unimodal order ideals. 8

22 References [] Joslyn, Cliff. Poset Ontologies and Concept Lattices as Semantic Hierarchies. Computer and Computational Sciences Los Alamos National Laboratory. [] Nvidia. The Benefits of Multiple CPU Cores in Mobile Devices. 00 [3] Leiserson, Charles E. et al. Parallel Computation of the Minimal Elements of a Poset. PASCO 0 Proceedings of the 4th International Workshop on Parallel and Symbolic Computation, [4] Stanley, Richard P. Chapter 3: Partially Ordered Sets. Enumerative Combinatorics. nd ed. Vol [5] Zeilberger, Doron. Kathy O Hara s Constructive Proof of the Unimodality of the Gaussian Polynomials. Thesis. Drexel University, 989. [6] Early, Edward Fielding. Chain and Antichain Enumeration in Posets, and B-ary Partitions. Thesis. Massachusetts Institute of Technology, 000. [7] Krattenthaler, Christian. The Number of Rhombus Tilings of a Symmetric Hexagon Which Contains a Fixed Rhombus on the Symmetry Axes, II. Thesis. Universität Wien, 000. [8] Proctor, Robert A. Solution of Two Difficult Combinatorial Problems with Linear Algebra. Thesis. University of California, Los Angeles, 98. The American Mathematical Monthly 89.0 (98): [9] Garg, Vijay K. Lattice Theory with Applications. Thesis. University of Texas at Austin, 997. [0] Levine, Lionel. Algebraic Combinatorics. Lecture 8. 0 March 0 [] Stembridge, John. Maple Packages for Symmetric Functions, Posets, Root Systems, and Finite Coxeter Groups. Computer software. Vers..4 9