We present a new construction of triple arrays by combining a symmetric 2-design with a resolution of another 2-design. This is the first general method capable of producing non-extremal triple arrays. We call the triple arrays which can be obtained in this way resolvable. We employ the construction to produce the first examples of (21 & times; 15, 63)-triple arrays, and enumerate all resolvable (7 & times; 15, 35)-triple arrays, of which there was previously only a single known example. An infinite subfamily of Paley triple arrays turns out to be resolvable. We also introduce a new intermediate object, unordered triple arrays, that are to triple arrays what symmetric 2-designs are to Youden rectangles, and propose a strengthening of Agrawal's long-standing conjecture on the existence of extremal triple arrays. For small parameters, we completely enumerate all unordered triple arrays, and use this data to corroborate the new conjecture. We construct several infinite families of resolvable unordered triple arrays, and, in particular, show that all ((q + 1) & times; q(2), q(q + 1))-triple arrays are resolvable and are in correspondence with finite affine planes of order q.

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares. We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of $6 \times 10$ triple arrays on $15$ symbols, $7 \times 8$ triple arrays on $14$ symbols and $5 \times 16$ triple arrays on $20$ symbols. Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.

We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui, and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and for some admissible parameter sets, we prove non-existence results. We give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, in particular, we prove constructively that AO-arrays exist for all feasible parameter sets. Finally, we investigate connections to Youden rectangles and binary pseudo-Youden designs.

We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and generate exhaustive lists of admissible parameters. For some admissible parameter sets, we prove non-existence results. We also give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, and investigate connections to Youden rectangles and binary pseud Youden designs.

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR \emph{homogeneous}. In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR \emph{transitive}. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix \emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

We investigate the rigidity properties of rod configurations . Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

In this paper we first study $k \times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call \emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.

This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss the rigidity properties of geometric realizations of incidence geometries in relation to the rigidity of geometric realizations of other well-known structures, such as graphs and hypergraphs.The $2$-plane matroid is also discussed. Further, we extend a result of Whiteley to determine necessary conditions for an incidence geometry of points and lines with exactly three points on each line, or 3-uniform hypergraphs, to have a minimally rigid realization as points and lines in the plane. We also give examples to show that these conditions are not sufficient. Finally, we examine the rigidity properties of $v_k$-configurations. We provide several examples of rigid $v_3$-configurations, and families of flexible geometric $v_3$-configurations. The exposition of the material is supported by many figures.

ver the course of the last few years, I have used in my teaching a variety of textbooks in the very broadly construed area of discrete mathematics.In one of these, the delightful Discrete Mathematics, by Norman L. Biggs, second edition [Big02], which I used in an introductory course in discrete mathematics, there is a section on how to introduce the natural numbers axiomatically, and in one subsection, the principle of mathematical induction is introduced as an axiom.This axiom is then used in a subsection on greatest and least members to prove that every nonempty subset of the natural numbers has a least member.The fact that the natural numbers have this property is usually called the well-ordering principle.It should be noted that sometimes this term is used for the well-ordering theorem, which states that every set can be well-ordered.It seemed to me that there was something missing from the exposition, namely a remark along the lines that ''we could have taken the well-ordering principle as an axiom instead of induction and still gotten the same structure, namely the familiar natural numbers.''True enough, when I went back to the first (revised) edition of Biggs's book [Big89], which I had read for the introductory discrete mathematics course I took as an undergraduate, I found that the well-ordering principle was taken as an axiom, and the induction principle was a theorem.This would seem to indicate that either way of presenting things would yield the same outcome.At this point, I decided to try to find a more solid, explicit, and detailed source for the equivalence of the induction principle and the well-ordering principle.So I searched for relevant articles in the MathSciNet database, and found the seemingly aptly titled article ''The equivalence of the multiplication, pigeonhole, induction, and well-ordering principles'' [HS88] and the related article ''Placing the pigeonhole principle within the defining axioms of the integers'' [HS80].Unexpectedly, what piqued my interest was not the articles themselves, but rather the reviews by Perry Smith.According to the Mathematics Genealogy Project database, Smith is an academic grandson of the famous logician Stephen Kleene (whose doctoral dissertation, incidentally, is titled A Theory of Positive Integers in Formal Logic), completing his PhD at UCLA in 1970 with a thesis entitled Some Contributions to Montague's Abstract Recursion Theory.Smith's publication record in MathSciNet comprises only two papers, but he has reviewed at least 169 papers in the database.From the review of the article mentioned first, what I took away was the remark that ''The authors work in Zermelo-Fraenkel set theory, but such arguments should be given in a weaker system of set theory or arithmetic in which the principles in question are not theorems' ' [Smi88].The other review deserves to be cited in full:The authors argue very informally that the pigeonhole principle can replace the induction axiom or the well-ordering principle in the set-theoretic characterization of the natural numbers.However, this claim must be formulated carefully if it is to be correct.For example, (1) the ordinals less than x þ x satisfy the first four (Dedekind-) Peano postulates and the well-ordering principle, but not the induction

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of $4 \times 8$ rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.

Mathematicians routinely pass judgements on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is, that the proof fits the theorem in an optimal way. It is also common to judge that one proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being more or less fitting, and provide examples from several different mathematical fields.

The zero forcing number of a graph is a graph parameter based on a color change process, which starts with a state, where all vertices are colored either black or white. In the next step a white vertex turns black, if it is the only white neighbor of some black vertex, and this step is then iterated. The zero forcing number Z(G) is defined as the minimum cardinality of a set S of black vertices such that the whole vertex set turns black.

The purpose of this note is to present an example of a proof by induction that in the opinion of the present author has great aesthetic value. The proof in question is Thomassen's proof that planar graphs are 5-choosable. I give a self-contained presentation of this result and its proof, and a personal account of why I think this proof is beautiful.A secondary purpose is to more widely publicize this gem, and hopefully make it part of a standard set of examples for examining characteristics of proofs by induction.

This paper addresses the question of when triple arrays can be constructed from Youden squares by removing a column together with the symbols therein, and then exchanging the role of columns and symbols. The scope of the investigation is limited to the standard case of triple arrays with \(v=r+c-1\). For triple arrays with \(\lambda _{cc}=1\) it is proven that they can never be constructed in this way, and for triple arrays with \(\lambda _{cc}=2\) it is proven that there always exists a suitable Youden square and a suitable column for this construction. Further, it is proven that Youden square constructed from a certain family of difference sets never give rise to triple arrays in this way but always gives rise to double arrays. Finally, it is proven that all triple arrays in the single known infinite family, the Paley triple arrays, can all be constructed in this way for some suitable choice of Youden square and column.

We discuss here the notion of mathematical fit, a concept that might relate to mathematical explanation and mathematical beauty. We specify two kinds of fit a proof can have, intrinsic and extrinsi ...

In the spirit of Ryser's theorem, we prove sufficient conditions on k, and m so that k xexm Latin boxes, i.e. partial Latin cubes whose filled cells form a k x x m rectangular box, can be extended to akxnxm latin box, and also to akxnxn latin box, where n is the number of symbols used, and likewise the order of the Latin cube.We also prove a partial Evans type result for Latin cubes, namely that any partial Latin cube of order n with at most n 1 filled cells is completable, given certain conditions on the spatial distribution of the filled cells.