Alexander Burstein

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Mobius function of an interval [@s,@p] in the poset of permutations ordered by pattern containment in the case where @p is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,...,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Mobius function in the case where @s and @p are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. We also show that the Mobius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Mobius function of an interval [@s,@p] of separable permutations is bounded by the number of occurrences of @s as a pattern in @p. Another consequence is that for any separable permutation @p the Mobius function of (1,@p) is either 0, 1 or -1.

Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.

On Unavoidable Sets of Word Patterns

We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern overlaps introduced in this paper, which is a subgraph of the de Bruijn graph and which we prove to be Hamiltonian. In other cases we reduce a problem under consideration to known facts on unavoidable sets of words. We also give a relation between our problem and the extensively studied universal cycles and prove that there exists a universal cycle for word patterns of any length over any alphabet. The Stirling numbers of the second kind and the Mobius function appear in our results.

Permutation Patterns: Restricted patience sorting and barred pattern avoidance

Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for permutations avoiding the barred (generalized) permutation pattern

Packing sets of patterns

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Combinatorial properties of permutation tableaux

. Motivated by this and a recently formulated geometric form for Patience Sorting in terms of certain intersecting lattice paths, we study the related themes of restricted input and avoidance of similar barred permutation patterns. One such result is to characterize those permutations for which Patience Sorting is an invertible algorithm as the set of permutations simultaneously avoiding the barred patterns

A lattice path approach to counting partitions with minimum rank t

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Words Restricted by Patterns with at Most 2 Distinct Letters

and

Counting occurrences of some subword patterns

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On Some Properties of Permutation Tableaux

. We then enumerate this avoidance set, which involves convolved Fibonacci numbers.