An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

Osculating Paths and Oscillating Tableaux

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths are permitted to share lattice points, but not to cross or share lattice edges. Such paths correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices and various subclasses thereof. Referring to points of the rectangle through which no or two paths pass as vacancies or osculations respectively, the case of primary interest is tuples of paths with a fixed number $l$ of vacancies and osculations. It is then shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for osculating paths involving sums over oscillating tableaux.Comment: 65 pages; expanded versio

Partially directed paths in a wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by $Y = \pm pX$, and an asymmetric wedge defined by the lines $Y= pX$ and Y=0, where $p > 0$ is an integer. We prove that the growth constant for all these models is equal to $1+\sqrt{2}$, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when $p=1$. From these we find asymptotic formulas for the number of partially directed paths of length $n$ in a wedge when $p=1$. The functional recurrences are solved by a variation of the kernel method, which we call the ``iterated kernel method''. This method appears to be similar to the obstinate kernel method used by Bousquet-Melou. This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi $\theta$-functions, and have natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT

Osculating paths and oscillating tableaux

The combinatorics of certain tuples of osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. The paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths within a tuple are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices. Of primary interest here are path tuples with a fixed number l of vacancies and osculations, where vacancies or osculations are points of the rectangle through which respectively no or two paths pass. It is shown that there exist natural bijections which map each such path tuple P to a pair (t,eta), where eta is an oscillating tableau of length l (i.e., a sequence of l+1 partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and t is a certain, compatible sequence of l weakly increasing positive integers. Furthermore, each vacancy or osculation of P corresponds to a partition in eta whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for tuples of osculating paths involving sums over oscillating tableaux

Symmetrized models of last passage percolation and non-intersecting lattice paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure

A generalization of Aztec diamond theorem, part I

We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic Combinatoric, 1992) by showing that the numbers of tilings of a certain family of regions in the square lattice with southwest-to-northeast diagonals drawn in are given by powers of 2. We present a proof for the generalization by using a bijection between domino tilings and non-intersecting lattice paths.Comment: 18 page

Growth models, random matrices and Painleve transcendents

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlev\'e II transcendent plays a prominent role.Comment: 27 pages, 5 figure

The representation of the symmetric group on m-Tamari intervals

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journa