Walks Models in the Quarter Plane

This is a the documentation that can be found inside the code of the package comb_walks.

This package offers a unified interface for working with generating functions and related object to Walks in the Quarter Plane. Assume we have a set

\[\mathcal{S} \subset \{-1,0,1\}^2 \setminus \{(0,0)\}\]

of, so called, valid steps. A walk according to \(\mathcal{S}\) is a series of points in \(\mathbb{Z}^2\) \(P_0=(0,0), P_1, \ldots, P_n\) such that for all \(k \in \{1,...,n\}\):

\[P_n - P_{n-1} \in \mathcal{S}.\]

We also say that the length of such walk is \(n\).

Let now be \(q_{i,j,k}^{\mathcal{S}}\) be the number of walks of length \(n\) and ends at \((i,j)\). Consider then the generating function:

\[Q_{\mathcal{S}}(x,y,t) = \sum_{i,j,k \geq 0} q_{i,j,k}^{\mathcal{S}} x^i y^j t^k \in \mathbb{Q}[x,y][[t]].\]

This package will allows the user to extract information of \(Q_{\mathcal{S}}\) providing an easy interface to create Walk models (see class WalkModel).

For using this package, use the import command:

from comb_walks import *

The use of this package on the main unweighted walks models can be seeing in the main webpage.

This code is under the terms of GNU General Public License version 3.

This package was partially funded by the Austrian Science Fund (FWF): W1214-N15, project DK15.

It was also supported in part by the ANR DeRerumNatura project, grant ANR-19-CE40-0018 of the French Agence Nationale de la Recherche.