Pattern Avoidability with Involution

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Anick-type resolutions and consecutive pattern avoidance

For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra considerations in the same sense as the corresponding inversion formula for avoiding word patterns comes from the well known Anick's resolution.Comment: 16 pages. Preliminary version, comments are welcom