Abstract
In this chapter, we define corridor paths, which are a type of lattice paths. We introduce the corridor and vertex numbers of a corridor, and provide motivation for the techniques of the next chapter.
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Notes
- 1.
For background and applications, see [24, 26, 43].
- 2.
A walk in a graph is defined as a finite sequence of adjacent vertices of the graph. Here, adjacent means connected by an edge. A path graph \(P_m\) is a graph on m vertices that are connected in a row by edges. A few helpful texts covering introductory graph theory include Brualdi [14], Epp [23], and Harris-Hirst-Mossinghoff [31], though we do not require an extensive understanding of graph theory for our purposes.
- 3.
I.e. the street along which he or she is walking is finite—this seems to me to be a much more reasonable scenario than the concept of the infinite street.
- 4.
An exhaustive bibliography of the work done in this field would be next to impossible to compile, but the interested reader may consult [7, 8, 12, 17, 18, 22, 25, 26, 33, 37, 38, 43, 45, 53, 56, 57].
- 5.
Our ideas are intimately related to the concept of reflectable walks, but more about this connection later.
- 6.
Alternatively, one could imagine a drunkard’s walk in which the drunkard frequently stops to think about which way to move next.
- 7.
OEIS sequence A001405.
- 8.
This type of extension by antisymmetry, prevalent throughout the signal processing literature, should not really be considered a trick, but rather a useful method to aid in analyzing certain kinds of sequences (see e.g., [21]).
- 9.
More recent exposition of the reflection principle is surveyed in [1, 17, 30, 37].
- 10.
However it would be interesting to consider three-way moves as well.
- 11.
A Laurent polynomial is a function of the form \(g(x) = \sum _{k=M}^{N} a_kx^k\), where \(M, N \in \mathbb {Z}\). In particular, negative powers are permitted.
- 12.
See also OEIS sequence A001764.
- 13.
- 14.
The reason that a spreadsheet program is especially useful is that the recursive formula only needs to be defined once. Then you can “drag” that formula up, down, and to the right to fill out the spreadsheet.
- 15.
See also [9].
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Ault, S., Kicey, C. (2019). Lattice Paths and Corridors. In: Counting Lattice Paths Using Fourier Methods. Applied and Numerical Harmonic Analysis(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26696-7_1
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