62 research outputs found
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Kronecker product identities from D-finite symmetric functions
Using an algorithm for computing the symmetric function Kronecker product of
D-finite symmetric functions we find some new Kronecker product identities. The
identities give closed form formulas for trace-like values of the Kronecker
product.Comment: 6 page
Two Non-holonomic Lattice Walks in the Quarter Plane
We present two classes of random walks restricted to the quarter plane whose
generating function is not holonomic. The non-holonomy is established using the
iterated kernel method, a recent variant of the kernel method. This adds
evidence to a recent conjecture on combinatorial properties of walks with
holonomic generating functions. The method also yields an asymptotic expression
for the number of walks of length n
Analytic aspects of the shuffle product
There exist very lucid explanations of the combinatorial origins of rational
and algebraic functions, in particular with respect to regular and context free
languages. In the search to understand how to extend these natural
correspondences, we find that the shuffle product models many key aspects of
D-finite generating functions, a class which contains algebraic. We consider
several different takes on the shuffle product, shuffle closure, and shuffle
grammars, and give explicit generating function consequences. In the process,
we define a grammar class that models D-finite generating functions
Regularity in Weighted Graphs a Symmetric Function Approach
This work describes how the class of k-regular multigraphs with edge multiplicities from a finite set can be expressed using symmetric species results of Mendez. Consequently, the generating functions can be computed systematically using the scalar product of symmetric functions. This gives conditions on when the classes are D-finite using criteria of Gessel, and a potential route to asymptotic enumeration formulas
- …