12 research outputs found

    A polyominoes-permutations injection and tree-like convex polyominoes

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    AbstractPlane polyominoes are edge-connected sets of cells on the orthogonal lattice Z2, considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [n], and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence an=6an−1−14an−2+16an−3−9an−4+2an−5, and compute the closed-form formula an=2n+2−(n3−n2+10n+4)/2

    Combinatorics of diagonally convex directed polyominoes

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    AbstractA new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney's generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q-enumeration of this object can be solved by an application of Gessel's q-analog of the Lagrange inversion formula

    Monotone Sequences in Combinatorial Structures

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    Symmetry of monotone sequences arise in many combinatorial structures, the classical examples being inversions and coinversions in permutations. Another example is crossings and nestings in matchings, partitions and permutations. These examples can be generalized to fillings of Ferrers diagrams and further generalized to moon polyominoes. This dissertation first introduces layer polyominoes, then extends the joint symmetry between northeast and southeast chains exhibited in moon polyominoes. For a given structure it’s not always true that symmetry of crossings and nestings holds. We introduce a type of matching, called an alternating matching, where the distribution of crossings and nestings is not symmetric. We prove a necessary and sufficient condition for an alternating matching to be non-nesting and use this to partially enumerate non-nesting alternating matchings. Finally, we prove several results on crossings and nestings in graphs. First we show that the crossing number and nesting numbers are unrelated, i.e. there are families of graphs with no crossings and with nestings numbers that diverge and vice versa. Second we give a bijection between plane trees and bi-colored motzkin paths. Lastly, we provide a generating function for a special class of Ferrers diagrams, where each row a fixed length shorter than the previous row, and the filling of the diagram has no southeast chains

    Asymptotic enumeration of non-crossing partitions on surfaces

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    We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft

    Algorithms Seminar, 2001-2002

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    These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Subject index volumes 1–92

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    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum
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