Vol. 10 no. 2

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1. Culminating paths

Mireille Bousquet-Mélou ; Yann Ponty.

Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not context-free. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating fucntion of culminating paths ending at a (generic) height k. In the case a=b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a>b, we obtain the asymptotic behaviour by a simpler argument. When a= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a

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2. On symmetric structures of order two

Michel Bousquet ; Cédric Lamathe.

Let (w_n)0 < n be the sequence known as Integer Sequence A047749 In this paper, we show that the integer w_n enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.

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3. A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata

David Callan.

We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant. We also give a formula for the number of acyclic automata with a given set of sources.

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4. On the size of induced acyclic subgraphs in random digraphs

Joel Spencer ; C.R. Subramanian.

Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that $mas(D) \le \frac{2}{ln(1-p)^-1} (ln np + 3e)$ almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) $mas(D) = \frac{2(ln np)}{ln(1-p)^-1} (1± o(1))$. This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n − 1) directed edges independently with probability p.

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5. Spanning forests on the Sierpinski gasket

Shu-Chiuan Chang ; Lung-Chi Chen.

We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.

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6. Leftmost derivations of propagating scattered context grammars: a new proof

Tomáš Masopust ; Jiří Techet.

In 1973, V. Virkkunen proved that propagating scattered context grammars which use leftmost derivations are as powerful as context-sensitive grammars. This paper brings a significantly simplified proof of this result.

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7. Analysis of some parameters for random nodes in priority trees

Alois Panholzer.

Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.

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