Coxeter-biCatalan combinatorics

We pose counting problems related to the various settings for Coxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to count "twin" pairs of objects from a corresponding problem in Coxeter-Catalan combinatorics. We show that the problems all have the same answer, and, for a given finite Coxeter group W, we call the common solution to these problems the W-biCatalan number. We compute the W-biCatalan number for all W and take the first steps in the study of Coxeter-biCatalan combinatorics.Comment: 53 pages, 8 figures. version2: Small expository changes to reflect the fact that "double-positive" Catalan polynomials have already appeared as the local h-polynomials of the positive cluster complex (Athanasiadis-Savvidou). version3: Expository changes reflecting referee suggestions. To appear in JAC

Chains in the noncrossing partition lattice

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice.Comment: Version 2: Several expository changes made, including changes in the abstract, thanks to helpful suggestions from several readers of Version 1. (See the Acknowledgments section of the paper.

2-neighborly 0/1-polytopes of dimension 7

We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors. It enables us to list some of their combinatorial properties. In particular, we have found a 2-neighborly polytope with 14 vertices and 16 facets.Comment: 8 pages, 1 figure, 6 table

A generalized Goulden-Jackson cluster method and lattice path enumeration

The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths---both with height bounded and unbounded---by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulae.Comment: 31 page

Pattern avoidance in labelled trees

We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words, permutations, coloured permutations etc. The notion of Wilf equivalence for patterns in permutations admits a straightforward generalisation for (sets of) tree patterns; we describe classes for trees with small numbers of leaves, and give several bijections between trees avoiding pattern sets from the same class. We also explain a few general results for tree pattern avoidance, both for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining the operadic contex

On floors and ceilings of the k-Catalan arrangement

The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fu{\ss}-Catalan number $Cat^{(k)}(\Phi)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.Comment: 12 page

Clusters, generating functions and asymptotics for consecutive patterns in permutations

We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns

Cataland: Why the Fuss?

The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.Comment: 132 pages, v2: major revisions and expansion, section on representation theory adde

Combinatorial Structure of the Deterministic Seriation Method with Multiple Subset Solutions

Seriation methods order a set of descriptions given some criterion (e.g., unimodality or minimum distance between similarity scores). Seriation is thus inherently a problem of finding the optimal solution among a set of permutations of objects. In this short technical note, we review the combinatorial structure of the classical seriation problem, which seeks a single solution out of a set of objects. We then extend those results to the iterative frequency seriation approach introduced by Lipo (1997), which finds optimal subsets of objects which each satisfy the unimodality criterion within each subset. The number of possible solutions across multiple solution subsets is larger than $n!$, which underscores the need to find new algorithms and heuristics to assist in the deterministic frequency seriation problem.Comment: 8 pages, 2 figure

Parallel Enumeration of Triangulations

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.Comment: 27 pages, 5 figure