Marni Mishna

Two non-holonomic lattice walks in the quarter plane

We present two classes of random walks restricted to the quarter plane with non-holonomic generating functions. The non-holonomicity is established using the iterated kernel method, a variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions [M. Mishna, Classifying lattice walks in the quarter plane, J. Combin. Theory Ser. A 116 (2009) 460-477]. The method also yields an asymptotic expression for the number of walks of length n.

Classifying lattice walks restricted to the quarter plane

This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of {0,+/-1}^2. New enumerative results are presented for several classes, each obtained with a variant of the kernel method.

Singularity analysis via the iterated kernel method

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

AVERAGE-CASE ANALYSIS OF PERFECT SORTING BY REVERSALS

A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NP-hard. Here we show that, despite this worst-case analysis, with probability one, sorting can be done in polynomial time. Further, we find asymptotic expressions for the average length and number of reversals in commuting permutations, an interesting sub-class of signed permutations.

Asymptotic Lattice Path Enumeration Using Diagonals

We consider d-dimensional lattice path models restricted to the first orthant whose defining step sets exhibit reflective symmetry across every axis. Given such a model, we provide explicit asymptotic enumerative formulas for the number of walks of a fixed length: the exponential growth is given by the number of distinct steps a model can take, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. The generating function of each model is first expressed as the diagonal of a multivariate rational function, then asymptotic expressions are derived by analyzing the singular variety of this rational function. Additionally, we show how to compute subdominant growth, reflect on the difference between rational diagonals and differential equations as data structures for D-finite functions, and show how to determine first order asymptotics for the subset of walks that start and end at the origin.

Effective scalar products of D-finite symmetric functions

Many combinatorial generatig functions can be expressed as combinatios of symmetric functions. or extracted as sub-series and specializations from such combinations. Gessel has outlined lagrge class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessels work by providing algorithms that compute differential equations, these generating functions satisfy in the case they are given a scalar product of symmetric functions in Gessels class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.

Enumeration of Cayley graphs and digraphs

This paper deals with the enumeration of various families of Cayley graphs and digraphs. Both the directed and undirected cases of the following three families are studied: Cayley graphs on cyclic groups, known as circulant graphs, of square-free order, circulant graphs of arbitrary order n whose connection sets are subsets of the group of units of Zn, and Cayley graphs on elementary abelian groups.

A combinatorial understanding of lattice path asymptotics

Abstract We provide a combinatorial derivation of the exponential growth constant for counting sequences of lattice path models restricted to the quarter plane. The values arise as bounds from analysis of related half planes models. We give explicit formulas, and the bounds are provably tight. The strategy is easily generalizable to cones in higher dimensions, and has implications for random generation.

Tableau sequences, open diagrams, and Baxter families

Walks on Youngs lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ź , end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram - open diagrams. Remarkably, two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case, and a generating function proof and new conjecture in the second case.

Set Partitions with No m-Nesting

A partition of \(\{1,\ldots,n\}\) has an m-nesting if it contains at least m disjoint blocks, and a subset of 2m points \(i_{1} < i_{2} <\ldots < i_{m} < j_{m} < j_{m-1} <\ldots < j_{1}\), such that i l and j l are in the same block for all 1 ≤ l ≤ m, but no other pairs are in the same block. In this note, we use generating trees to construct the class of partitions with no m-nesting, determine functional equations satisfied by the associated generating functions, and generate enumerative data for m ≥ 4.