Abstract
Following the work of Cano and Díaz, 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.
Highlights
Cano and Dıaz [1, 2] have recently explored a novel method of obtaining continuous analogues of discrete objects such as binomial coefficients and Catalan numbers
They realized these discrete quantities as the number of certain lattice paths. They considered directed paths as continuous extensions of lattice paths and define moduli spaces of directed paths. They declared the volumes of these moduli spaces to be the continuous versions of the original discrete objects
We extend some of Cano and Dıaz’s work to higher dimensions, obtaining a partial differential equation that the continuous multinomials satisfy, which generalizes the electronic journal of combinatorics 26(3) (2019), #P3.57 the partial differential equation of Cano and Dıaz from dimension 2 to dimension n
Summary
Cano and Dıaz [1, 2] have recently explored a novel method of obtaining continuous analogues of discrete objects such as binomial coefficients and Catalan numbers. Following Cano and Dıaz, we define a directed path using the same set-up as in definition (1) above except for the important difference that each coefficient λi is a non-negative real number. The moduli space of all directed paths from the origin to q ∈ Rd is defined by the union of the following polytopes: This moduli space can be endowed with a natural flat metric, which is one of the innovations in the work of Cano and Dıaz. There was a min√or error in their calculation of P (q, c) amounting to an extra multiplicative factor of d, where d is the dimension of the ambient space
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
