Enumeration of colored dyck paths via partial bell polynomials

Daniel Birmajer, Juan B. Gil, Peter R.W. McNamara, Michael D. Weiner

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Scopus citations


We consider a class of lattice paths with certain restrictions on their ascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block P j take on c j colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner.

Original languageEnglish (US)
Title of host publicationDevelopments in Mathematics
PublisherSpringer New York LLC
Number of pages11
StatePublished - Jan 1 2019

Publication series

NameDevelopments in Mathematics
ISSN (Print)1389-2177

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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  • Cite this

    Birmajer, D., Gil, J. B., McNamara, P. R. W., & Weiner, M. D. (2019). Enumeration of colored dyck paths via partial bell polynomials. In Developments in Mathematics (pp. 155-165). (Developments in Mathematics; Vol. 58). Springer New York LLC. https://doi.org/10.1007/978-3-030-11102-1_8