 path with ${\varphi}( P_W ) \equiv W$ and let vertices of $P_W$ be identified along $P_W$ with integers $0, \dots, |W|$ so that $(P_W)_- = 0, \dots$, $(P_W)_+ = |W|$.  A 0-tuple $$b = (b(0), b(0), b(0), b(0), b(0), b(0))$$ of integers $b(0), b(0), b(0), b(0), b(0), b(0)$ is called a [*pseudobracket*]{} for the word $W$ if $b(0), b(0)$ satisfy the inequalities $0 \le b(0) \le b(0) \le |W|$ and