Inclusion Matrices and Chains

Given integers $t$, $k$, and $v$ such that $0\leq t\leq k\leq v$, let $W_{tk}(v)$ be the inclusion matrix of $t$-subsets vs. $k$-subsets of a $v$-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset $2^{[v]}$ into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by $W_{\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ is obtained as well as a new proof of the well known theorem on the necessary and sufficient conditions for existence of integral solutions of the system $W_{tk}\bf{x}=\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$.Comment: Accepted for publication in Journal of Combinatorial Theory, Series

On set systems with restricted intersections modulo p and p-ary t-designs

We consider bounds on the size of families ℱ of subsets of a v-set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections. We improve the known bound when ℱ is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families ℱ holds with equality, then ℱ is the set of blocks of what we call a p-ary t-design for certain values of t. This motivates us to make a few observations about p-ary t-designs for their own sake

A New Proof of a Classical Theorem in Design Theory

AbstractWe present a new proof of the well known theorem on the existence of signed (integral) t-designs due to Wilson and Graver and Jurkat

Constructing Regular Self-complementary Uniform Hypergraphs

AMS Subject Classication Codes: 05C65, 05B05 05E20, 05C85.In this paper, we examine the possible orders of t-subset-regular self-complementary k-uniform hypergraphs, which form examples of large sets of two isomorphic t-designs. We reformulate Khosrovshahi and Tayfeh-Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1-subset-regular self-complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1.https://onlinelibrary.wiley.com/doi/abs/10.1002/jcd.2028

The design of mixture experiments in the presence of covariates

Over the past three decades, the design and analysis of mixture experiments has been an active area of research, often driven by industrial applications. However, the construction of block designs for mixture experiments and trend-free orderings of the mixtures are problems that have been largely ignored until recently. These two problems form the principal subjects of this dissertation after presenting some key concepts in the design and analysis of mixture experiments;Block designs are constructed using combinatorial structures called symbolic and integral mixture mates of strength t. Certain pairs of Latin squares are a special case of symbolic mixture mates. One flexible method of constructing integral mixture mates of strength t uses the theory of trade-off for m-ary designs. In addition to mixture mates, block designs may be constructed via other methods. When the region of interest is a constrained subregion of the simplex, confounding in fractional factorial designs or asymmetrical orthogonal arrays may be used to produce orthogonal block designs. Methods for constructing non-orthogonal block designs utilizing factorial designs or orthogonal arrays in another manner are given. Finally, we formulate algorithms that allocate a given set of mixtures to blocks in such a way that an objective function is maximized;Trend-free mixture orderings allow uncorrelated estimators of mixture model parameters and deterministic trend parameters to be obtained. Deterministic trends may be induced by time effects or other lurking variables. Given a trend-free ordering of a factorial design in p - 1 factors, we illustrate how a trend-free order of mixtures can be found by transforming the p - 1 factors into p mixture variables, using one of the many transformations available, see Cornell (1991) for example. If the experimental region is a constrained subregion of the simplex, trend-free run orders are constructed using trend-free factorial designs in p - 1 factors as a template and incorporating a p-th factor by adjusting the levels of the other p - 1 components for each row. Nearly trend-free mixture orders are also found by ordering the mixtures according to an objective function

Large Sets of t-Designs

We investigate the existence of large sets of t-designs. We introduce t-wise equivalence and (n, t)-partitionable sets. We propose a general approach to construct large sets of t-designs. Then, we consider large sets of a prescribed size n. We partition the set of all k-subsets of a v-set into several parts, each can be written as product of two trivial designs. Utilizing these partitions we develop some recursive methods to construct large sets of t-designs. Then, we direct our attention to the large sets of prime size. We prove two extension theorems for these large sets. These theorems are the only known recursive constructions for large sets which do not put any additional restriction on the parameters, and work for all t and k. One of them, has even a further advantage; it increase the strength of the large set by one, and it can be used recursively which makes it one of a kind. Then applying this theorem recursively, we construct large sets of t-designs for all t and some blocksizes k. Hartman conjectured that the necessary conditions for the existence of a large set of size two are also sufficient. We suggest a recursive approach to the Hartman conjecture, which reduces this conjecture to the case that the blocksize is a power of two, and the order is very small. Utilizing this approach, we prove the Hartman conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely many k, and for the rest of them there are at most k/2 exceptions. In Chapter 4 we consider the case k = t + 1. We modify the recursive methods developed by Teirlinck, and then we construct some new infinite families of large sets of t-designs (for all t), some of them are the smallest known large sets. We also prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.</p

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe

Self-Complementary Hypergraphs

In this thesis, we survey the current research into self-complementary hypergraphs, and present several new results. We characterize the cycle type of the permutations on n elements with order equal to a power of 2 which are k-complementing. The k-complementing permutations map the edges of a k-uniform hypergraph to the edges of its complement. This yields a test to determine whether a finite permutation is a k-complementing permutation, and an algorithm for generating all self-complementary k-uniform hypergraphs of order n, up to isomorphism, for feasible n. We also obtain an alternative description of the known necessary and sufficient conditions on the order of a self-complementary k-uniform hypergraph in terms of the binary representation of k. We examine the orders of t-subset-regular self-complementary uniform hyper- graphs. These form examples of large sets of two isomorphic t-designs. We restate the known necessary conditions on the order of these structures in terms of the binary representation of the rank k, and we construct 1-subset-regular self-complementary uniform hypergraphs to prove that these necessary conditions are sufficient for all ranks k in the case where t = 1. We construct vertex transitive self-complementary k-hypergraphs of order n for all integers n which satisfy the known necessary conditions due to Potocnik and Sajna, and consequently prove that these necessary conditions are also sufficient. We also generalize Potocnik and Sajna's necessary conditions on the order of a vertex transitive self-complementary uniform hypergraph for certain ranks k to give neces- sary conditions on the order of these structures when they are t-fold-transitive. In addition, we use Burnside's characterization of transitive groups of prime degree to determine the group of automorphisms and antimorphisms of certain vertex transitive self-complementary k-uniform hypergraphs of prime order, and we present an algorithm to generate all such hypergraphs. Finally, we examine the orders of self-complementary non-uniform hypergraphs, including the cases where these structures are t-subset-regular or t-fold-transitive. We find necessary conditions on the order of these structures, and we present constructions to show that in certain cases these necessary conditions are sufficient.University of OttawaDoctor of Philosophy in Mathematic

A new basis for trades

Research supported in part by a grant from the Research Council of the University of TehranSIGLEITItal