id	author	title	date	pages	extension	mime	words	sentences	flesch	summary	cache	txt
work_2mdy3t4xane2hdgqagdmp6jlju	Dan Hirsch	More ties than we thought	2015	15	.pdf	application/pdf	6846	1135	50	Keywords Necktie knots, Formal language, Automata, Chomsky hierarchy, Generating functions and Fink & Mao (1999) defined a formal language for describing tie-knots, encoding the this language to enumerate all tie-knots that could reasonably be tied with a normal-sized Fink and Mao, together with an analysis of the asymptotic complexity class of the tie-knots A tie-knot has to be tied by winding and tucking one of the two blades around the other: move to tuck the blade U nder the tie itself.2 The notation proposed by Fink & Mao (2000) We can write a context-free grammar for the arbitrary depth tuck tie-knots. of generality, we can assume that a tie-knot starts by putting the active blade in region R: For our necktie-knot grammars, the sizes are the winding lengths of the ties, and it is • The generating function for Fink and Mao necktie-knots is • The generating function for single tuck necktie-knots is	./cache/work_2mdy3t4xane2hdgqagdmp6jlju.pdf	./txt/work_2mdy3t4xane2hdgqagdmp6jlju.txt