RO-TST 2013 Problem 17

Given a positive integer $n$ , consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$ . The entries of the array are all either $0$ or $1$ , and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$ , and $a_{ij}$ is $1$ otherwise. Let $S$ denote the set of binary sequences of length $n$ , and define a map $f \colon S \to S$ via $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$ . Determine the number of fixed points of $f$ .