3,049 research outputs found

    Introduction to Partially Ordered Patterns

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    We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution. Moreover, we discuss q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea

    Generalized pattern avoidance with additional restrictions

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    Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an (n1)(n-1)-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a generalized 3-pattern with no dashes and whose k leftmost or k rightmost letters form either an increasing or decreasing subword. Moreover, we find a bijection between n-permutations that avoid the pattern 132 and begin with the pattern 12 and increasing rooted trimmed trees with n+1 nodes.Comment: 18 page

    On graphs with representation number 3

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    A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if (x,y)(x,y) is an edge in EE. A graph is word-representable if and only if it is kk-word-representable for some kk, that is, if there exists a word containing kk copies of each letter that represents the graph. Also, being kk-word-representable implies being (k+1)(k+1)-word-representable. The minimum kk such that a word-representable graph is kk-word-representable, is called graph's representation number. Graphs with representation number 1 are complete graphs, while graphs with representation number 2 are circle graphs. The only fact known before this paper on the class of graphs with representation number 3, denoted by R3\mathcal{R}_3, is that the Petersen graph and triangular prism belong to this class. In this paper, we show that any prism belongs to R3\mathcal{R}_3, and that two particular operations of extending graphs preserve the property of being in R3\mathcal{R}_3. Further, we show that R3\mathcal{R}_3 is not included in a class of cc-colorable graphs for a constant cc. To this end, we extend three known results related to operations on graphs. We also show that ladder graphs used in the study of prisms are 22-word-representable, and thus each ladder graph is a circle graph. Finally, we discuss kk-word-representing comparability graphs via consideration of crown graphs, where we state some problems for further research

    The sigma-sequence and counting occurrences of some patterns, subsequences and subwords

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    We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.Comment: 10 page

    Quantum measurements and the Abelian Stabilizer Problem

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    We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.Comment: 22 pages, LATE