Sequences with changing dependencies
Consider words over an alphabet with n letters. Fisher [Amer. Math. Monthly, 96 (1989), pp. 610-614] calculated the number of distinct words of length l assuming certain pairs of letters commute. In this paper we are interested in a more general setting where the pairs of letters that commute at a certain position of a word depend on the initial segment of the word. In particular, we show that if for each word at each position any letter fails to commute with at most a constant number of other letters, then the number of distinct words of length l is at most Cn+l for some constant C. We use this result to obtain a lower bound on the number of diagonal flips required in the worst case to transform one n-vertex labeled triangulated planar graph into some other one. This has previously been proved in [D. D. Sleator, R. E. Tarjan, and W. P. Thurston, SIAM J. Discrete Math., 5 (1992), pp. 428-450] by different methods.
SIAM Journal on Discrete Mathematics
Balister, P., Bollobás, B., & Gerke, S. (2008). Sequences with changing dependencies. SIAM Journal on Discrete Mathematics, 22 (3), 1149-1154. https://doi.org/10.1137/060663611