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The Identity Transform of a Permutation and its Applications

Frosini A., Battaglino D., Rinaldi S., Socci S.

Fundamenta Informaticae

2015

Vol. 141, nr 2/3

191--205

Starting from a Theorem by Hall, we define the identity transform of a permutation π as C(π) = (0 + π(0), 1 + π(1), ..., (n - 1) + π(n - 1)), and we define the set Cn = {(C(π) : π ∈ Sn}, where Sn is the set of permutations of the elements of the cyclic group Zn. In the first part of this paper we study the set Cn: we show some closure properties of this set, and then provide some of its combinatorial and algebraic characterizations and connections with other combinatorial structures. In the second part of the paper, we use some of the combinatorial properties we have determined to provide a different algorithm for the proof of Hall's Theorem.

Polygons Drawn from Permutations

Bilotta S., Rinaldi S., Socci S.

Fundamenta Informaticae

2013

Vol. 125, nr 3/4

329--342

In this paper we consider the class of column-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations (π1, π2). First, using a geometric construction, we prove that for every permutation π there is at least one column-convex permutomino P such that π1(P) = π or π2(P) = π. In the second part of the paper, we show how, for any given permutation π, it is possible to define a set of logical implications F(p) on the points of π, and prove that there exists a column-convex permutomino P such that π1(P) = π if and only if F(p) is satisfiable. This property can be then used to give a characterization of the set of column-convex permutominoes P such that π1(P) = π.