Question 53
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 $i \gt 1$ 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 $f: S \to S$ via $f: (a_{11}, a_{12}, \dots, a_{1n}) \mapsto (a_{n1}, a_{n-1,2}, \dots, a_{1n})$. Determine the number of fixed points of $f$.
Geoffrey Smith, United Kingdom, RMM 2013 Short List
Let $S$ denote the set of binary sequences of length $n$, and define a map $f: S \to S$ via $f: (a_{11}, a_{12}, \dots, a_{1n}) \mapsto (a_{n1}, a_{n-1,2}, \dots, a_{1n})$. Determine the number of fixed points of $f$.
Geoffrey Smith, United Kingdom, RMM 2013 Short List