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  • 1.
    Andren, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed)
    Abstract [en]

    We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

  • 2.
    Andrén, Lina J.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Casselgren, Carl Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

  • 3.
    Björklund, Johanna
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Simulation relations for pattern matching in directed graphs2013In: Theoretical Computer Science, ISSN 0304-3975, E-ISSN 1879-2294, Vol. 485, p. 1-15Article in journal (Refereed)
    Abstract [en]

    We consider the problem of finding the occurrences of a pattern tree t in a directed graph g, and propose two algorithms, one for preprocessing and one for searching for t in g. It is assumed that the object graph itself is large and static, and that the pattern tree is small and frequently updated. To model varying abstraction levels in the data, we work with partially ordered alphabets and compute simulation relations rather than equivalence relations. In particular, vertices and edges are labelled with elements from a pair of preorders instead of unstructured alphabets. Under the above assumptions, we obtain a search algorithm that runs in time O(height (t) . vertical bar t vertical bar . vertical bar(V-g(+/-)t/R-g(+/-)t vertical bar(2)) where vertical bar (V-g(+/-)t/R-g(+/-)t)vertical bar is the number of equivalence classes in the coarsest simulation relation R-g(+/-)t on the graph g((+/-))t, the disjoint union of g and t. This means that the size of the object graph only affects the running time of the search algorithm indirectly, because of the groundwork done by the preprocessing routine in time O(k . vertical bar g vertical bar . vertical bar(V-g/R-g)vertical bar(2)), where vertical bar(V-g/R-g) is the number of equivalence classes in the coarsest simulation relation R-g on g, taking k = vertical bar V-g vertical bar(2) in the general case and k = height (g) if g is acyclic.

  • 4. Cavenagh, Nicholas J.
    et al.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Partial latin squares are avoidable2006Report (Other academic)
  • 5. Cutler, Jonathan
    et al.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Latin squares with forbidden entries2004Report (Other academic)
  • 6. Cutler, Jonathan
    et al.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Latin squares with forbidden entries2006In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 13, no 1, p. R47-Article in journal (Refereed)
  • 7.
    Denley, Tristan
    et al.
    Austin Peay State University.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Extending partial latin cubes2014In: Ars combinatoria, ISSN 0381-7032, Vol. 113, p. 405-414Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Johansson, Robert
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Introduktion till högre studier i matematik2012 (ed. 1)Book (Other academic)
  • 9.
    Jäger, Gerold
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shcherbak, Denys
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Enumeration of t-tuples of Mutually Orthogonal Latin Rectangles and Finite GeometriesManuscript (preprint) (Other academic)
  • 10.
    Jäger, Gerold
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shcherbak, Denys
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Triples of Orthogonal Latin and Youden Rectangles of small order2019In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 27, no 4, p. 229-250Article in journal (Refereed)
    Abstract [en]

    We have performed a complete enumeration of non-isotopic triples of mutually orthogonal k × n Latin rectangles for k ≤ n ≤ 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×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 × 8 rectangles, formed by juxtaposing two   selected copies of complete sets of MOLS of order 4.

  • 11.
    Jäger, Gerold
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Shcherbak, Denys
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Enumeration of Youden Rectangles of Small ParametersManuscript (preprint) (Other academic)
  • 12.
    Markström, Klas
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Unavoidable arrays2010In: Contributions to Discrete Mathematics, ISSN 1715-0868, E-ISSN 1715-0868, Vol. 5, no 1, p. 90-106Article in journal (Refereed)
    Abstract [en]

    An n x n array is avoidable if for each set of n symbols there is a Latin square on these symbols which diers from the array in every cell. We characterise all unavoidable square arrays with at most 2 symbols, and all unavoidable arrays of order at most 4. We also identify a number of general families of unavoidable arrays, which we conjecture to be a complete account of unavoidable arrays. Next, we investigate arrays with multiple entries in each cell, and identify a number of families of unavoidable multiple entry arrays. We also discuss fractional Latin squares, and their connections to unavoidable arrays.

    We note that when rephrasing our results as edge list-colourings of complete bipartite graphs, we have a situation where the lists of available colours are shorter than the length guaranteed by Galvin's Theorem to allow proper colourings.

  • 13.
    Nilson, Tomas
    et al.
    Mittuniversitetet.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Triple arrays and Youden squares2015In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 75, no 3, p. 429-451Article in journal (Refereed)
    Abstract [en]

    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.

  • 14.
    Raman Sundström, Manya
    et al.
    Umeå University, Faculty of Science and Technology.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Beauty as fit: A metaphor in mathematics?2013In: Research in Mathematics Education, Vol. 15, no 2, p. 199-200Article in journal (Refereed)
    Abstract [en]

    Beauty, which plays a central role in the practice of mathematics, is almost absent in discussions of school mathematics. This is problematic, because students will decide whether or not to continue their studies inmathematics without having an accurate picture of what the subject is about. In order to have a discussion about how to introduce beauty into the school mathematicscurriculum, we need to have a clear idea about what beauty means. That is the aim ofthis study, with a focus on characterising beauty in mathematical proof.

  • 15.
    Raman Sundström, Manya
    et al.
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Mathematical fit: A first approximation2015In: Proceedings of the ninth conference of the Euorpean Society for research in Mathermatics education (CERME9) / [ed] Krainer, K Vondrova, N, Prague: Charles University , 2015, p. 185-191Conference paper (Refereed)
    Abstract [en]

    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 extrinsic, and provide characteristics that help distinguish different proofs of the same theorem.

  • 16.
    Raman Sundström, Manya
    et al.
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC). Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Two beautiful proofs of Pick's theorem2011In: Proceedings of Seventh Congress of European Research in Mathematics Education, 2011Conference paper (Refereed)
    Abstract [en]

    We present two different proofs of Pick’s theorem and analyse in what ways might be perceived as beautiful by working mathematicians. In particular, we discuss two concepts, generality and specificity, that appear to contribute to beauty in different ways. We also discuss possible implications on insight into the nature of beauty in mathematics, and how the teaching of mathematics could be impacted, especially in countries in which discussions of beauty and aesthetics are notably absent from curricular documents.

  • 17.
    Raman-Sundström, Manya
    et al.
    Umeå University, Faculty of Science and Technology, Department of Science and Mathematics Education.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Mathematical fit: a case study2018In: Philosophia mathematica, ISSN 0031-8019, E-ISSN 1744-6406, Vol. 26, no 2, p. 184-210Article in journal (Refereed)
    Abstract [en]

    Mathematicians routinely pass judgments 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 theoremin an optimal way. It is also common to judge that a 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 full text will be freely available from 2020-06-01 00:00
  • 18.
    Raman-Sundström, Manya
    et al.
    Umeå University, Faculty of Science and Technology, Umeå Mathematics Education Research Centre (UMERC).
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sinclair, Nathalie
    Faculty of Education, Simon Fraser University, CANADA.
    The Nature and Experience of Mathematical Beauty2016In: Journal of Humanistic Mathematics, ISSN 2159-8118, E-ISSN 2159-8118, Vol. 6, no 1, p. 3-7Article in journal (Refereed)
  • 19.
    Shcherbak, Denys
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Jäger, Gerold
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Zero Forcing Number of Bijection Graphs2015In: Proceedings of 26th International Workshop om Combinatorial Algorithms (IWOCA 2015), Berlin-Heidelberg: Springer, 2015, Vol. 9538, p. 334-345Conference paper (Refereed)
    Abstract [en]

    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.

    In this paper we study Z(G) for the class of bijection graphs, where a bijection graph is a graph on 2n vertices that can be partitioned into two parts with n vertices each, joined by a perfect matching. For this class of graphs we show an upper bound for the zero forcing number and classify the graphs that attain this bound. We improve the general lower bound for the zero forcing number, which is Z(G)&#x2265;&#x03B4;(G)" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">Z(G)≥δ(G)Z(G)≥δ(G), for certain bijection graphs and use this improved bound to find the exact value of the zero forcing number for these graphs. This extends and strengthens results of Yi (2012) about the more restricted class of so called permutation graphs.

  • 20.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A Beautiful Proof by Induction2016In: Journal of Humanistic Mathematics, ISSN 2159-8118, E-ISSN 2159-8118, Vol. 6, no 1, p. 73-85Article in journal (Refereed)
    Abstract [en]

    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.

  • 21.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A Note on Completing Partial Latin Squares2009In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 45, p. 117-124Article in journal (Refereed)
  • 22.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Latin Squares with Prescriptions and Restrictions2011In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 51, p. 77-87Article in journal (Refereed)
  • 23.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the intricacy of avoiding multiple-entry arrays2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 20, p. 3030-3036Article in journal (Refereed)
    Abstract [en]

    Let A be any n x n array on the symbols vertical bar n vertical bar = {1, . . . , n}, with at most in symbols in each cell. An n x n Latin square L on the symbols till is said to avoid A if no entry in L is present in the corresponding cell of A, and A is said to be avoidable if such a Latin square L exists. The intricacy of this problem is defined to be the minimum number of arrays into which A must be split in order to ensure that each part is avoidable. We present lower and upper bounds for the intricacy, and conjecture that the lower bound is in fact the correct answer. (C) 2012 Elsevier B.V. All rights reserved.

  • 24.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Partial latin squares are avoidable2011In: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 15, no 3, p. 485-497Article in journal (Refereed)
    Abstract [en]

    A square array is avoidable if for each set of n symbols there is an n x n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m >= 2 any partial Latin square of order 4m - 1 is avoidable, thus concluding the proof that any partial Latin square of order at least 4 is avoidable.

  • 25.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The intricacy of avoiding arrays2005Report (Other academic)
  • 26.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The intricacy of avoiding arrays2005Manuscript (preprint) (Other academic)
  • 27.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The intricacy of avoiding arrays is 22006In: Discrete Mathematics, ISSN 0012-365X, Vol. 306, p. 531-532Article in journal (Refereed)
1 - 27 of 27
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