Geodesic
Counting lattice paths by using difference equations with non-constant coefficients
The Bulletin of Irkutsk State University. Series Mathematics, Tome 44 (2023), pp. 55-70 Cet article a éte moissonné depuis la source Math-Net.Ru

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The lattice paths can be counted by the virtue of their step vectors that are aligned to the positive octant. A path can go from one point to an infinite others if there is no restriction applied such that each point only has finitely many predecessors. The linear difference equations with non-constant coefficients will be utilised to incorporate this restriction to study lattice paths that lie on or over a line having a rational slope. The generating functions are obtained and is based on developing a specific method to compute the number of restricted lattice paths.
Keywords: generating function, difference equation, functional equation, lattice path. 
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Sreelatha Chandragiri. Counting lattice paths by using difference equations with non-constant coefficients. The Bulletin of Irkutsk State University. Series Mathematics, Tome 44 (2023), pp. 55-70. http://geodesic.mathdoc.fr/item/IIGUM_2023_44_a4/

[1] Abramov S.A., Barkatou M.A, Van Hoeij M., Petkovsek M., “Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences”, Journal of Symbolic Computation, 46:11 (2011), 1205–1228 | DOI

[2] Banderier C., Flajolet P., “Basic analytic combinatorics of directed lattice paths”, Theoretical Computer Science, 281 (2002), 37–80 | DOI

[3] Bloom D.M., “Singles in a sequence of coin tosses”, The College Mathematics Journal, 29:2 (1998), 120–127 | DOI

[4] Bousquet-M$\acute{e}$lou M., Petkov$\check{s}$ek M., “Linear recurrences with constant coefficients: the multivariate case”, Discrete Mathematics, 225 (2000), 51–75 | DOI

[5] Chandragiri S., “Difference Equations and Generating Functions for Some Lattice Path Problems”, Journal of Siberian Federal University. Mathematics Physics, 12:5 (2019), 551–559 | DOI

[6] Duchon P., “On the Enumeration and Generation of Generalized Dyck Words”, Discrete Mathematics, 225 (2000), 121–135 | DOI

[7] Gelfond A.O., The calculus of finite differences, KomKniga Publ, M., 2006 (in Russian)

[8] Gessel I.M., “A Factorization for Formal Laurent Series and Lattice Path Enumeration”, Journal of Combinatorial Theory, Series A, 28:3 (1980), 321–337 | DOI

[9] Labelle J., Yeh Y.N., “Generalized Dyck paths”, Discrete Mathematics, 82:1 (1990), 1–6 | DOI

[10] Leinartas E.K., “Multiple Laurent series and fundamental solutions of linear difference equations”, Siberian Mathematical Journal, 48:2 (2007), 268–272 | DOI

[11] Lyapin A.P., Chandragiri S., “Generating Functions for Vector Partitions and a Basic Recurrence Relation”, Journal of Difference Equations and Applications, 25:7 (2019), 1052–1061 | DOI

[12] Lyapin A.P., Chandragiri S., “The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones”, Journal of Siberian Federal University. Mathematics Physics, 13:2 (2020), 187–196 | DOI

[13] de Moivre A., “De Fractionibus Algebraicis Radicalitate Immunibus ad Fractiones Simpliciores Reducendis, Deque Summandis Terminis Quarumdam Serierum Aequali Intervallo a Se Distantibus”, Philosophical Transactions, 32:373 (1722), 162–178 | DOI

[14] Stanley R., Enumerative Combinatorics, v. 1, Cambridge University Press, 1990