Bijections on two variations of noncrossing partitions

We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schröder paths and Schröder paths without peaks at even height.     

A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.     

A bijection between ordered trees and 2-Motzkin paths and its many consequences

A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0,1,2}-trees, Schröder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport-Schinzel sequences.RésuméUne nouvelle bijection entre arbres ordonnés et chemins de Motzkin bicolorés est présentée, avec ses nombreuses conséquences en ce qui concerne les arbres ordonnés ainsi que d'autres structures combinatoires telles que chemins de Dyck, buissons, arbres de type {0,1,2}, chemins de Schröder, structures secondaires de type RNA, partitions non croisées, chemins de Fine, et enfin suites de Davenport-Schinzel.     

Linked partitions and linked cycles

The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the n-th large Schröder number r_n, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, k-Stirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs.     

Limits of areas under lattice paths

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays.     

A Distributive Lattice Structure Connecting Dyck Paths,Noncrossing Partitions and 312-avoiding Permutations

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.     

Pairs of noncrossing free Dyck paths and noncrossing partitions

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n+1] with n+1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.     

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.     

Riordan paths and derangements

Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and -avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

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.     

Some Combinatorics Related to Central Binomial Coefficients: Grand-Dyck Paths, Coloured Noncrossing Partitions and Signed Pattern Avoiding Permutations

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.     

Refined Chung-Feller theorems for lattice paths

In this paper we prove a strengthening of the classical Chung-Feller theorem and a weighted version for Schröder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating Schröder paths with flaws.     

2-Binary trees: Bijections and related issues

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain statistics.     

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Johansson.     

Pattern Avoidance in Alternating Sign Matrices

We generalize the definition of a pattern from permutations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schr?der numbers, 1, 2, 6, 22, 90, 394, .... We give a bijection between 132-avoiding alternating sign matrices and Schr?der paths, which gives a refined enumeration. We also show that the 132-, 123-avoiding alternating sign matrices are counted by every second Fibonacci number. Received January 2, 2007     

The butterfly decomposition of plane trees

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem as well as some generalizations and variations follow quickly from the butterfly decomposition. We next obtain two involutions on free Dyck paths and free Schröder paths, leading to parity results and combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of Klazar's generating function for the number of chains in plane trees. Finally we study the total size of chains in plane trees with n edges and show that the average size of such chains tends asymptotically to (n+9)/6.     

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.     

Counting Dyck Paths by Area and Rank

The set of Dyck paths of length 2n inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: area (the area under the path) and rank (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting (q, t)-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: One refines an identity of Carlitz and Riordan; the other refines the notion of γ-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the n-cycle (12· · ·n) in the absolute order on the symmetric group.     

Inversion polynomials for 321-avoiding permutations

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formula for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.