“Deco” polyominoes, permutations and random generation

Polyominoes determined by permutations

In the plane Z × Z a cell is a unit square, and a polyomino is a finite connected union of cells having no cut point. Polyominoes are defined up to translations. A column (row) of a polyomino is the intersection between the polyomino and an infinite strip of cells whose centers lie on a vertical (horizontal) line. The enumeration problem for general polyominoes is difficult to solve and still o...

Random Generation of hv-Convex Polyominoes with Given Horizontal Projection

Random and exhaustive generation of permutations and cycles

In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well known. We consider both methods in a unified way, and discuss their relation with exhaustive generation methods. We analyse several random variables associat...

Consecutive Patterns: From Permutations to Column-Convex Polyominoes and Back

We expose the ties between the consecutive pattern enumeration problems as­ sociated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column­ convex polyominoes, and of words to be applied directly to the enumeration of per­ mutations by consecutive patterns. We deduce a host of new consecutive patt...

Random Walks and Random Permutations

A connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE. The scaling of the distribution gives...