On how many Fibonacci words is the Plancherel measure concentrated

Source abstract

On the set $\mathbb{YF}_n$ of words in $1$ and $2$ with digit sum $n$, we consider the probability distribution $μ_P$, corresponding to the Plancherel measure on paths in the Young--Fibonacci graph. We study the asymptotics as $n$ grows of the entropy of the distribution $μ_P$ and of the minimum number of words on which $μ_P$ is almost entirely concentrated.