On the Number of Broken Derived Terms of a Rational Expression
Pierre-Yves Angrand, Sylvain Lombardy, Jacques Sakarovitch · Journal of automata, languages and combinatorics · 2010
Bounds are given on the number of broken derived terms (a variant of Antimirov's "partial derivatives") of a rational expression E. It is shown that this number is less than or equal to $2\ell(E) + 1$ in the general case, where $\ell(E)$ is the literal length of the expression $E$, and that the classical bound $\ell(E) + 1$ which holds for partial derivatives also holds for broken derived terms if $E$ is in star normal form. In a second part of the paper, the influence of the bracketing of an expression on the number of its derived terms is also discussed.