A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

The NP-complete Permutation Pattern Matching problem asks whether a $k$-permutation $P$ is contained in a $n$-permutation $T$ as a pattern. This is the case if there exists an order-preserving embedding of $P$ into $T$. In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of $\mathcal{O}(1.79^{\mathsf{run}(T)}\cdot n\cdot k)$, where $\mathsf{run}(T)$ denotes the number of alternating runs of $T$. This algorithm is particularly well-suited for instances where $T$ has few runs, i.e., few ups and downs. Moreover, since $\mathsf{run}(T)<n$, this can be seen as a $\mathcal{O}(1.79^{n}\cdot n\cdot k)$ algorithm which is the first to beat the exponential $2^n$ runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixed-parameter tractability result is not possible for $\mathsf{run}(P)$

Off-diagonal ordered Ramsey numbers of matchings

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the "off-diagonal" ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random matchings with interval chromatic number $2$.Comment: 15 pages, 3 figure

Bijections from weighted Dyck paths to Schroeder paths

Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schroeder paths, and from a similar set of Dyck paths to big Schroeder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with 231-avoiding permutations.Comment: 16 pages, TikZ figures, includes Sage cod

Pattern avoidance in matchings and partitions

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of B\'ona for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin--West--Xin and Jel\'{\i}nek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.Comment: 34 pages, 12 Figures, 5 Table

Edge Effects on Local Statistics in Lattice Dimers: A Study of the Aztec Diamond (Finite Case)

We compute the probability of any local pattern at an arbitrary position in a random dimer configuration in a square grid with an Aztec-diamond boundary.Comment: 41 pages, 28 figure

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

A Bijection for Crossings and Nestings

For a subclass of matchings, set partitions, and permutations, we describe a direct bijection involving only arc annotated diagrams that not only interchanges maximum nesting and crossing numbers, but also all refinements of crossing and nesting numbers. Furthermore, we show that the bijection cannot be applied to a similar class of coloured arc annotated diagrams

Splittings and Ramsey Properties of Permutation Classes

We say that a permutation p is 'merged' from permutations q and r, if we can color the elements of p red and blue so that the red elements are order-isomorphic to q and the blue ones to r. A 'permutation class' is a set of permutations closed under taking subpermutations. A permutation class C is 'splittable' if it has two proper subclasses A and B such that every element of C can be obtained by merging an element of A with an element of B. Several recent papers use splittability as a tool in deriving enumerative results for specific permutation classes. The goal of this paper is to study splittability systematically. As our main results, we show that if q is a sum-decomposable permutation of order at least four, then the class Av(q) of all q-avoiding permutations is splittable, while if q is a simple permutation, then Av(q) is unsplittable. We also show that there is a close connection between splittings of certain permutation classes and colorings of circle graphs of bounded clique size. Indeed, our splittability results can be interpreted as a generalization of a theorem of Gy\'arf\'as stating that circle graphs of bounded clique size have bounded chromatic number.Comment: 34 pages, 6 figure

A general theory of Wilf-equivalence for Catalan structures

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences, or generating functions, of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal classes defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal classes have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of a given size and show that it is exponentially smaller than the corresponding Catalan number. In other words these "coincidental" equalities are in fact very common among principal classes. Our results also allow us to prove, in a unified and bijective manner, several known Wilf-equivalences from the literature.Comment: 24page

Crossings and alignments of permutations

We derive the continued fraction form of the generating function of some new $q$-analogs of the Eulerian numbers $\hat{E}_{k,n}(q)$ introduced by Lauren Williams building on work of Alexander Postnikov. They are related to the number of alignments and weak exceedances of permutations. We show how these numbers are related to crossing and generalized patterns of permutations We generalize to the case of decorated permutations. Finally we show how these numbers appear naturally in the stationary distribution of the ASEP model.Comment: 11 pages, 2 figure