Permutominoes are polyominoes defined by suitable pairs of permutations. In this paper we provide a formula to count the number of convex permutominoes of given perimeter. To this aim we define the transform of a generic pair of permutations, we characterize the transform of any pair defining a convex permutomino, and we solve the counting problem in the transformed space.

Abstract. Straight-Line Programs (SLP) are widely used compressed representations of words. In this work we study the rational transformations and the literal shuffle of words compressed via SLP, proving that the first preserves the compression rate, while the second does not. As a consequence, we prove a tight bound for the descriptional complexity of 2D texts compressed via SLP. Finally, we observe that the Pattern Matching Problem for texts expressed by the literal shuffle of compressed words is NP-complete. However, we present a parameter-tractable algorithm for this problem, working in polynomial time whenever the length of the pattern is polynomially related to that of the text.

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model IPM k (n), a generalization of the sand pile model SPM(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice IPM k (n): this lets us design an algorithm which generates all the ice piles of IPM k (n) in amortized time O(1) and in space O(√ n).

We study the problem of testing whether a context-free language is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid. As an application, we obtain a polynomial time algorithm for testing if a context-free language is included in a Dyck language (the best previous algorithm for this problem was doubly exponential).

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