Simultaneous avoidance of generalized patterns

In [BabStein] Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case. Also we consider permutations that avoid a pattern of the form x - yz or xy - z and begin with one of the patterns 12 . . . k, k(k -1). . .1, 23. . . k1, (k - 1)(k - 2). . . 1k or end with one of the patterns 12. . .k, k(k - 1). . .1, 1k(k - 1). . .2, k12. . . (k - 1). For each of these cases we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides we generalize some of the obtained results as well as some of the results given in [Kits]: we consider permutations avoiding certain generalized 3-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.