  Consider the following one-player game. Take a well-formed sequence of
opening and closing brackets. As a move, the player can pair any opening
bracket with any closing bracket to its right, erasing them. The goal is to
re-pair (erase) the entire sequence, and the complexity of a strategy is
measured by its width: the maximum number of nonempty segments of symbols
(separated by blank space) seen during the play.
  For various initial sequences, we prove upper and lower bounds on the minimum
width sufficient for re-pairing. (In particular, the sequence associated with
the complete binary tree of height $n$ admits a strategy of width
sub-exponential in $\log n$.) Our two key contributions are (1) lower bounds on
the width and (2) their application in automata theory: quasi-polynomial lower
bounds on the translation from one-counter automata to Parikh-equivalent
nondeterministic finite automata. The latter result answers a question by Atig
et al. (2016).
