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Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.

The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set of explicit families.

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We show that the left-greedy algorithm is a better algorithm than the right-greedy algorithm for sorting permutations using t stacks in series when t > 1. We also supply a method for constructing some permutations that can be sorted by t stacks in series and from this get a lower bound on the number of permutations of length n that are sortable by t stacks in series. Finally we show that the left-greedy algorithm is neither optimal nor defines a closed class of permutations for t > 2.

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation π to be k-pass sortable if π is sortable using k passes through the stack. Permutations that are 1-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of 2-pass sortable permutations in terms of their basis. We also show all k-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer k. Finally, we define the notion of tier of a permutation π to be the minimum number of passes after the first pass required to sort π. We then give a bijection between the class of permutations of tier t and a collection of integer sequences studied by Parker [16]. This gives an exact enumeration of tier t permutations of a given length and thus an exact enumeration for the class of (t + 1)-pass sortable permutations. Finally, we give a new derivation for the generating function in [16] and an explicit formula for the coefficients.

a b s t r a c tWe investigate the notion of almost avoiding a permutation: π almost avoids β if one can remove a single entry from π to obtain a β-avoiding permutation.

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