Symmetric Permutations Avoiding Two Patterns

Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D ₄, the symmetry group of a square. We examine pattern-avoiding permutations with 180° rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, triangular numbers, and powers of two.     

Restricted signed permutations counted by the Schröder numbers

Gire, West, and Kremer have found ten classes of restricted permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. In this paper we enumerate eleven classes of restricted signed permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. We obtain five of these enumerations by elementary methods, five by displaying isomorphisms with the classical Schröder generating tree, and one by giving an isomorphism with a new Schröder generating tree. When combined with a result of Egge and a computer search, this completes the classification of restricted signed permutations counted by the large Schröder numbers in which the set of restrictions consists of two patterns of length 2 and two of length 3.     

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.     

Restricted Simsun Permutations

A permutation is simsun if for all k, the subword of the one-line notation consisting of the k smallest entries does not have three consecutive decreasing elements. Simsun permutations were introduced by Simion and Sundaram, who showed that they are counted by the Euler numbers. In this paper we enumerate simsun permutations avoiding a pattern or a set of patterns of length 3. The results involve Motkzin, Fibonacci, and secondary structure numbers. The techniques in the proofs include generating functions, bijections into lattice paths and generating trees.     

Separable d-Permutations and Guillotine Partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula, and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between separable d-dimensional permutations and guillotine partitions of a 2 ^d-1-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain several results on patterns in permutations.     

A test for conjugacy of partitions of finite sets

In this note we present a simple test, involving a sequence of integers, which assures the conjugacy of a given partition P of a finite set S when our operations lead only to nonnegative integers. If negative integers appear in our operations, our test is inconclusive. The test, when conclusive, and an elementary property of permutations determine a conjugate for P.     

Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of ²n−1 permutations that pairwise generate the symmetric group Sn. There is no set of ²n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of ²n−2 even permutations that pairwise generate the alternating group An. There is no set of ²n−2+1 permutations having this property.     

Some combinatorial interpretations ofq-analogs of Schröder numbers

We introduce two definitions of Schröder numberq-analogs and show some combinatorial interpretations of theseq-numbers. We use the following combinatorial objects for these interpretations: Schröder paths, 1-colored parallelogram polyominoes and permutations with forbidden subsequences (4231, 4132). We enumerate these objects according to various parameters by means of a recentq-counting technique. We prove that the firstq-Schröder number enumerates of Schröder paths with respect to area and the number of permutation inversions, while the second one counts the 1-colored parallelogram polyominoes according to their width and area. Finally, we illustrate some relations among the parameters characterizing the combinatorial objects.     

ECO:a methodology for the enumeration of combinatorial objects

In this Paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. The basic idea of this method is the following: by means of an operator that performs a "local expansion" on the objects, we give some recursive constructions of these classes. We use these constructions to deduce some new funtional equations verified by classes' generating functions. By solving the functional equations, we enumerate the combinatorial objects according to various parameters. We show some applications of the method referring to some classical combinatorial objects, such as: trees, paths, polyminoes and permutations     

Restricted Dumont Permutations

We analyze the structure and enumerate Dumont permutations of the first and second kinds avoiding certain patterns or sets of patterns of length 3 and 4. Some cardinalities are given by Catalan numbers, powers of 2, little Schr?der numbers, and other known or related sequences. Received June 21, 2004     

Restricted Symmetric Permutations

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D ₈, the symmetry group of a square. Three other nontrivial subgroups of D ₈ also have invariant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S ₃. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations. Received September 14, 2006     

Permutations Sortable by n – 4 Passes Through a Stack

We characterise and enumerate permutations that are sortable by n – 4 passes through a stack. We conjecture the number of permutations sortable by n – 5 passes, and also the form of a formula for the general case n – k, which involves a polynomial expression.     

Counting with Borel’s triangle

Borel’s triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel’s triangle. We present various combinatorial interpretations of Borel’s triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.     

On convex permutations

A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.     

Periods in strings

In this paper we explore the notion of periods of a string. A period can be thought of as a shift that causes the string to match over itself. We obtain two sets of necessary and sufficient conditions for a set of integers to be the set of periods of some string (what we call the correlation of the string). We show that the number of distinct correlations of length n is independent of the alphabet size and is of order n^logn. By using generating function methods we enumerate the strings having a given correlation, and investigate certain related questions.     

Enumerating rc-Invariant Permutations with No Long Decreasing Subsequences

We use the Robinson-Schensted-Knuth correspondence and Schützenberger’s evacuation of standard tableaux to enumerate permutations and involutions which are invariant under the reverse-complement map and which have no decreasing subsequences of length k. These enumerations are in terms of numbers of permutations with no decreasing subsequences of length approximately \frack2;{{\frac{k}{2}};} we use known results concerning these quantities to give explicit formulas when k ≤ 6.     

Intersection theorems in permutation groups

The Hamming distance between two permutations of a finite setX is the number of elements ofX on which they differ. In the first part of this paper, we consider bounds for the cardinality of a subset (or subgroup) of a permutation groupP onX with prescribed distances between its elements. In the second part. We consider similar results for sets ofs-tuples of permutations; the role of Hamming distance is played by the number of elements ofX on which, for somei, the ith permutations of the two tuples differ.     

The r-cubical Lattice and a Generalization of the cd-index

In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtill's work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all ther_i's are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.     

Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

Stack words stem from studies on stack-sortable permutations and represent classical combinatorial objects such as standard Young tableaux, permutations with forbidden sequences and planar maps. We extend existing enumerative results on stack words and we also obtain new results. In particular, we make a correspondence between nonseparable 3×n rectangular standard Young tableaux (or stack words where elements satisfy a ‘Towers of Hanoi’ condition) and nonseparable cubic rooted planar maps with 2n vertices enumerated by 2ⁿ(3n)!/((2n+1)!(n+1)!). Moreover, these tableaux without two consecutive integers in the same row are in bijection with nonseparable rooted planar maps with n+1 edges enumerated by 2(3n)!/((2n+1)!(n+1)!).