Three Generalizations of Davenport-Schinzel Sequences

We present new, and mostly sharp, bounds on the maximum length of certain generalizations of Davenport-Schinzel sequences. Among the results are sharp bounds on order-$s$ {\em double DS} sequences, for all $s$, sharp bounds on sequences avoiding {\em catenated permutations} (aka formation free sequences), and new lower bounds on sequences avoiding {\em zig-zagging} patterns