Elena Barcucci

Reconstructing convex polyominoes from horizontal and vertical projections

In [1], we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections. We defined an algorithm that establishes the existence of a convex polyomino Λ whose horizontal and vertical projections are equal to a pair of assigned vectors (H,V), with H ∈ ℕ m and V ∈ ℕ n . Its computational cost is O(n4m4). In this paper, we introduce some operations for recontructing convex polyominoes by means of vectors Hs and Vs partial sums. These operations allows us to define a new algorithm whose complexity is less than O(n2m2).

Random generation of trees and other combinatorial objects

Abstract In this paper, we present a general method for the random generation of some classes of combinatorial objects. Our basic idea is to translate ECO method (Enumerating Combinatorial Objects) from a method for the enumeration of combinatorial objects into a random generation method. The algorithms we illustrate are based on the concepts of succession rule and generating tree: the former is a law that predicts the combinatorial object class growth according to a given parameter. The generating tree related to a given succession rule is a particular labelled plane tree that represents the rule in an extensive way. Each node of a generating tree can also be seen as a particular combinatorial object and so a random path in the generating tree coincides with the random generation of that combinatorial object. The generation is uniform if we take the probability of each branch to be selected into account when the path is generated. We also give the formulae evaluating complexity. Finally, we take the class of m -ary trees into consideration in order to illustrate our general method. In this case, the average time complexity of the generating algorithm can be estimated as O( mn ).

Exhaustive generation of combinatorial objects by ECO

Abstract.The problem of exhaustively generating combinatorial objects can currently be applied to many disciplines, such as biology, chemistry, medicine and computer science. A well known approach to the exhaustive generation problem is given by the Gray code scheme for listing n-bit binary numbers in such a way that successive numbers differ in exactly one bit position. In this work, we introduce an exhaustive generation algorithm, which is general for the classes of succession rules considered in [1]. We also show that our algorithm is efficient in an amortized sense; it actually uses only a constant amount of computation per object.

Medians of polyominoes: A property for reconstruction

In a previous report, we studied the problem of reconstructing a discrete set 𝒮 from its horizontal and vertical projections. We defined an algorithm that decides whether there is a convex polyomino 𝒮 whose horizontal and vertical projections are given by (H, V), with H ∈ ℕm and V ∈ ℕn. If there is at least one convex polyomino with these projections, the algorithm reconstructs one of them in O(n4m4) time. In this article, we introduce the geometrical concept of a discrete sets medians. Starting out from this geometric property, we define some operations for reconstructing convex polyominoes from their projections (H, V). We are therefore able to define a new algorithm whose complexity is less than O(n2m2). Hence, this algorithm is much faster than the previous one. At the moment, however, we only have experimental evidence that this algorithm decides if there is a convex polyomino whose projections are equal to (H, V), for all (H, V) instances.

The random generation of directed animals

Abstract In this paper, we propose an algorithm to randomly generate a directed animal. Directed animals are well-known combinatorial objects and have been widely used for modelling the physical phenomenon of percolation. The algorithm consists of three steps, and we prove that each of them is performed in linear time. Finally, we report the results of our experiments made by means of appropriate computer programs in order to give empirical evidence that our algorithm really works.

X-rays characterizing some classes of discrete sets

In this paper, we study the problem of determining discrete sets by means of their X-rays. An X-ray of a discrete set F in a direction u counts the number of points in F on each line parallel to u. A class F of discrete sets is characterized by the set U of directions if each element in F is determined by its X-rays in the directions of U. By using the concept of switching component introduced by Chang and Ryser [Comm. ACM 14 (1971) 21; Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14, The Mathematical Association of America, Rahway, 1963] and extended in [Discrete Comput. Geom. 5 (1990) 223], we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions. Gardner and Gritzmann [Trans. Amer. Math. Soc. 349 (1997) 2271] show that any set U of four directions having cross ratio that does not belong to {4/3,3/2,2,3,4}, characterizes the class of convex sets. We prove the converse, that is, if Us cross ratio is in {4/3,3/2,2,3,4}, then the hv-convex sets cannot be characterized by U. We show that if the horizontal and vertical directions do not belong to U, Gardner and Gritzmanns result cannot be extended to hv-convex polyominoes. If the horizontal and vertical directions belong to U and Us cross ratio is not in {4/3,3/2,2,3,4}, we believe that U characterizes the class of hv-convex polyominoes. We give experimental evidence to support our conjecture. Moreover, we prove that there is no number δ such that, if |U|⩾δ, then U characterizes the hv-convex polyominoes. This number exists for convex sets and is equal to 7 (see [Trans. Amer. Math. Soc. 349 (1997) 2271]).

From Motzkin to Catalan permutations

For every integer j>1, we dene a class of permutations in terms of certain forbidden subsequences. For j=1, the corresponding permutations are counted by the Motzkin numbers, and for j= 1 (dened in the text), they are counted by the Catalan numbers. Each value of j>1 gives rise to a counting sequence that lies between the Motzkin and the Catalan numbers. We compute the generating function associated to these permutations according to several parameters. For every j>1, we show that only this generating function is algebraic according to the length of the permutations. c 2000 Elsevier Science B.V. All rights reserved.

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math.217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.

Some permutations with forbidden subsequences and their inversion number

A permutation π avoids the subpattern τ iff π has no subsequence having all the same pairwise comparisons as τ, and we write π∈S(τ). We examine the classes of permutations, S(321),S(321,3142) and S(4231,4132), enumerated, respectively by the famous Catalan, Motzkin and Schroder number sequences. We determine their generating functions according to their length, number of active sites and inversion number. We also find the average inversion number for each class. Finally, we describe some bijections between these classes of permutations and some classes of parallelogram polyominoes, from which we deduce some relations between the parameters of Motzkin and Schroder permutations.

Some more properties of Catalan numbers

We want to illustrate some correspondences between Catalan numbers and combinatoric objects, such as plane walks, binary trees and some particular words. By means of under-diagonal walks, we give a combinatorial interpretation of the formula Cn = 1n+12nn defining Catalan numbers. These numbers also enumerate both words in a particular language defined on a four character alphabet and the corresponding walks made up of four different types of steps. We illustrate a bijection between n-long words in this language and binary trees having n + 1 nodes, after which we give a simple proof of Touchards formula.