45 research outputs found

    On the Factorization of Graphs with Exactly One Vertex of Infinite Degree

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    AbstractWe give a necessary and sufficient condition for the existence of a 1-factor in graphs with exactly one vertex of infinite degree

    Reduction of mm-Regular Noncrossing Partitions

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    In this paper, we present a reduction algorithm which transforms mm-regular partitions of [n]={1,2,...,n}[n]=\{1, 2, ..., n\} to (m1)(m-1)-regular partitions of [n1][n-1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers

    Multivariate Fuss-Catalan numbers

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    Catalan numbers C(n)=1n+1(2nn)C(n)=\frac{1}{n+1}{2n\choose n} enumerate binary trees and Dyck paths. The distribution of paths with respect to their number kk of factors is given by ballot numbers B(n,k)=nkn+k(n+kn)B(n,k)=\frac{n-k}{n+k}{n+k\choose n}. These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is surprising that the extension of this construction to 3 dimensions generates integers B3(n,k,l)B_3(n,k,l) that give a 2-parameter distribution of C3(n)=12n+1(3nn)C_3(n)=\frac 1 {2n+1} {3n\choose n}, which may be called order-3 Fuss-Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers B3(n,k,l)B_3(n,k,l). We obtain an explicit formula and a description in terms of trees and paths. Finally, we extend our construction to pp-dimensional arrays, and in this case we obtain a (p1)(p-1)-parameter distribution of Cp(n)=1(p1)n+1(pnn)C_p(n)=\frac 1 {(p-1)n+1} {pn\choose n}, the number of pp-ary trees

    Author index volume 43 (1983)

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    The Bohnenblust–Spitzer Algorithm and its Applications

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    The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, variances, and concentration inequalities for several random variables associated with a random walk including the number of vertices and length of the convex minorant, concave majorant, and convex hull

    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

    The Distribution of Patterns in Random Trees

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    Let T_nT\_n denote the set of unrooted labeled trees of size nn and let T_nT\_n be a particular (finite, unlabeled) tree. Assuming that every tree of T_nT\_n is equally likely, it is shown that the limiting distribution as nn goes to infinity of the number of occurrences of MM as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to μn\mu n and σ2n\sigma^2n, respectively, where the constants μ>0\mu>0 and σ0\sigma\ge 0 are computable
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