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A continuous analogue of lattice path enumeration. (English) Zbl 1420.05080

Summary: Following the work of L. Cano and R. Diaz [“Continuous analogues for the binomial coefficients and the Catalan numbers”, Preprint, arXiv:1602.09132], we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.

MSC:

05C30 Enumeration in graph theory
05C38 Paths and cycles
68R15 Combinatorics on words
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References:

[1] L. Cano and R. D´ıaz. Indirect Influences on Directed Manifolds. Advanced Studies in Contemporary Mathematics, 28-1, 93-114, 2018. · Zbl 1412.94258
[2] L. Cano and R. D´ıaz. Continuous analogues for the binomial coefficients and the Catalan numbers.arXiv:1602.09132v4. · Zbl 1524.05007
[3] P. Flajolet, R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. · Zbl 1165.05001
[4] I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products. 7th ed., Academic Press, San Diego, 2007.
[5] Y. Karshon, S. Sternberg and J. Weitsman. Exact Euler-Maclaurin formulas for simple lattice polytopes. Advances in Applied Mathematics, Volume 39, 2007, 1-50. · Zbl 1153.65006
[6] K. Knauer, L. Martnez-Sandoval and J. Luis Ram‘ırez. On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial, Discrete and Computational Geometry, October 2018, Volume 60, Issue 3, 698-719. the electronic journal of combinatorics 26(3) (2019), #P3.5712 · Zbl 1494.52014
[7] A. D. Kolesnik.Moment analysis of the telegraph random process, Buletinul Academiei De Stiint¸e a Republicii Moldova. Matematica, Number 1(68), 2012, 90- 107. · Zbl 1272.60064
[8] A. P. Prudnikov, Y. A. Brychkov and O.I. Marichev. Integrals and Series, Volume 2, Gordon and Breach Science Publishers, 1986. · Zbl 0606.33001
[9] W.T. Ross and H.S. Shapiro. Generalized Analytic Continuation, A.M.S., University Lecture Series 25, 2002. · Zbl 1009.30002
[10] T. Wakhare and C. Vignat. A continuous analog of lattice path enumeration: Part II, Online Journal of Analytic Combinatorics, 14, 2019. the electronic journal of combinatorics 26(3) (2019), #P3.5713 · Zbl 1427.05012
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