<abstract xmlns="http://www.w3.org/1999/xhtml">Permutations are frequently used in solving the genome rearrangement problem, whose goal is finding the shortest sequence of mutations transforming one genome into another. We introduce the Deletion-Insertion model (DI) to model small-scale mutations in species with linear chromosomes, such as humans. Applying one restriction to this model, we obtain the transposition model for genome rearrangement, which was shown to be NP-hard in [4]. We use combinatorial reasoning and permutation statistics to develop a polynomial-time algorithm to approximate the minimum number of transpositions required in the transposition model and to analyze the sharpness of several bounds on transpositions between genomes.</abstract>

Department of Mathematics, Rose-Hulman Institute of Technology

The Deletion-Insertion model applied to the genome rearrangement problem

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper basically completes a project to enumerate permutations avoiding a triple <italic>T</italic> of 4-letter patterns, in the sense of classical pattern avoidance, for every <italic>T</italic>. There are 317 symmetry classes of such triples <italic>T</italic> and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.</abstract>

Enumeration of permutations avoiding a triple of 4-letter patterns is almost all done

<abstract xmlns="http://www.w3.org/1999/xhtml">A graph <italic>G</italic> = (<italic>V;E</italic>) is word-representable if there is a word <italic>w</italic> over the alphabet <italic>V</italic> such that <italic>x</italic> and <italic>y</italic> alternate in <italic>w</italic> if and only if the edge (<italic>x; y</italic>) is in <italic>G</italic>. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not.There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed.It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].</abstract>

Colourability and word-representability of near-triangulations

<abstract xmlns="http://www.w3.org/1999/xhtml">Let λ be a partition of the positive integer <italic>n</italic> chosen uniformly at random among all such partitions. Let <italic>Ln</italic> = <italic>L</italic><italic>n</italic>(λ) and <italic>M</italic><italic>n</italic> = <italic>M</italic><italic>n</italic>(λ) be the largest part size and its multiplicity, respectively. For large <italic>n</italic>, we focus on a comparison between the partition statistics <italic>L</italic><italic>n</italic> and <italic>L</italic><italic>n</italic><italic>M</italic><italic>n</italic>. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of <italic>L</italic><italic>n</italic><italic>M</italic><italic>n</italic> – <italic>L</italic><italic>n</italic> grows as fast as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2015-0033_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>log</mo><mi>n</mi></mrow></math><tex-math>{1 \over 2}\log n</tex-math></alternatives></inline-formula>. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.</abstract>

On the largest part size and its multiplicity of a random integer partition

<abstract xmlns="http://www.w3.org/1999/xhtml">We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a <italic>k</italic>-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a <italic>k</italic>-ary language. The corresponding class of <italic>n</italic>-letter <italic>k</italic>-ary words is seen to be equinumerous with the closed walks of length <italic>n</italic> − 1 on the complete graph for <italic>k</italic> vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the <italic>k</italic>-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.</abstract>

Generalized Jacobsthal numbers and restricted <italic>k</italic>-ary words

<abstract xmlns="http://www.w3.org/1999/xhtml">A generalized (resp. <italic>p</italic>-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than <italic>p</italic>) where in any of its prefixes each positive integer <italic>i</italic> occurs at most as often as any integer less than <italic>i</italic>. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and <italic>p</italic>-ary ballot sequences when <italic>p</italic> is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.</abstract>

LIB, Université Bourgogne Franche-Comté,

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

AHEAD OF PRINT

Volume 30 (2022): Issue 4 (December 2022)

Volume 30 (2022): Issue 3 (October 2022) Special issue dedicated to 70th anniversary of prof. Peter Tallos

Volume 30 (2022): Issue 2 (August 2022)

Volume 30 (2022): Issue 1 (June 2022)

Volume 29 (2020): Issue 1 (December 2020)

Volume 28 (2019): Issue 1 (October 2019)

Volume 27 (2018): Issue 2 (August 2018)

Volume 27 (2018): Issue 1 (July 2018)

Volume 26 (2017): Issue 1 (June 2017)

Volume 25 (2015): Issue 2 (December 2015)

Volume 25 (2015): Issue 1 (September 2015)

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