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.


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