Abstract
In 2011, Beaton et al. analytically proved that the number of directed column-convex permutominoes of size $n$ is given by $(n+1)!/2$. In this paper, we provide a different proof of this statement using a bijective method. More precisely, we present a bijective correspondence between the class $D_n$ of directed column-convex permutominoes of size $n$ and a set of permutations (called $dcc$-permutations) of length $n+1$, which we prove to be counted by $(n+1)!/2$. The class of $dcc$-permutations is a new class of permutations counted by half factorial numbers, and here we show some combinatorial characterizations of this class, using the concept of logical formulas determined by a permutation and the notion of mesh pattern.
Highlights
Let P be a polyomino without “holes”, i.e. a polyomino whose boundary is a single loop, and having n−1 rows and n−1 columns, n 2; we assume without loss of generality that the south-west corner of its minimal bounding rectangle is placed at (1, 1)
The class of dcc-permutations is a new class of permutations counted by half factorial numbers, and here we show some combinatorial characterizations of this class, using the concept of logical formulas determined by a permutation and the notion of mesh pattern
We succeed in giving an alternative characterization of dcc-permutations, using the concept of logical formulas determined by a permutation, recently introduced in [4]
Summary
Let P be a polyomino without “holes”, i.e. a polyomino whose boundary is a single loop, and having n−1 rows and n−1 columns, n 2; we assume without loss of generality that the south-west corner of its minimal bounding rectangle is placed at (1, 1). Permutominoes was obtained in [2], where the authors determined a direct recursive construction for the column-convex permutominoes of a given size, leading to a functional equation. In this paper we consider the class Dn of directed column-convex permutominoes of size n 1. The authors of [2] analytically proved that the number of directed column-convex permutominoes of size n is (n + 1)!/2. We succeed in giving an alternative characterization of dcc-permutations, using the concept of logical formulas determined by a permutation, recently introduced in [4] Thank to this result, we are able to characterize the set dcc-permutations as a class of permutations avoiding two mesh patterns [7]
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