Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3-patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.

normality of statistics on permutation tableaux

Abstract. We find exact formulas and/or generating functions for the number of words avoiding 3-letter generalized multipermutation patterns and find which of them are equally avoided. 1.

Abstract. Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1, 2) or (2, 1). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns. 1.

Abstract. We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrímsson and Williams [9], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. 1.

Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 12-3-...-k (respectively; 21-3-...-k) in π. In the present note, we study the distribution of the statistics ekπ and fkπ in a permutation avoiding the classical pattern 1-3-2. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern 1-3-2 according to a given statistics on ekπ, or on fkπ. 1.

In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let fτ;r(n) be the number of 1-3-2-avoiding permutations on n letters that contain exactly r occurrences of τ, where τ a generalized pattern on k letters. Let Fτ;r(x) and Fτ(x, y) be the generating functions defined by Fτ;r(x) = � n n≥0 fτ;r(n)x and Fτ(x, y) = � r≥0 Fτ;r(x)y r. We find an explicit expression for Fτ(x, y) in the form of a continued fraction for where τ given as a generalized pattern; τ = 12-3-...-k, τ = 21-3-...-k, τ = 123...k, or τ = k...321. In particularly, we find Fτ(x, y) for any τ generalized pattern of length 3. This allows us to express Fτ;r(x) via Chebyshev polynomials of the second kind, and continued fractions. 1.

A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.

Abstract. We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern. 1.