Bootstrap percolation, the Schro¨der numbers, and the N -kings problem

A percolation process on $n \times n$ 0-1matrices is defined. This process is defined so that a zero entry becomes one if two or more of its neighbors have the value one. Entries that have the value one never change. The process halts when no more entries can change. The initial matrices are taken to be all the $n \times n$ permutation matrices.It is shown that the number of matrices that eventually become all ones is given by the Schroder numbers. Asymptotically, the proportion of such matrices approaches zero. Next, matrices that exhibit no growth at all are considered. The number of such matrices is given in terms of a generating function, and the proportion of such matrices approaches $e^{ - 2} $ as n goes to infinity. The methods used involve bracketing, trees, and generating functions.