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

Abstract

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
Pages155-165
Number of pages11
DOIs
StatePublished - Jan 1 2019

Publication series

NameDevelopments in Mathematics
Volume58
ISSN (Print)1389-2177

Fingerprint

Bell Polynomials
Dyck Paths
Enumeration
Lattice Paths
Partial
Building Blocks
Ascent
Recurrence relation
Colouring
Convolution
Simplify
Count
Restriction

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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
Birmajer, Daniel ; Gil, Juan B. ; McNamara, Peter R.W. ; Weiner, Michael D. / Enumeration of colored dyck paths via partial bell polynomials. Developments in Mathematics. Springer New York LLC, 2019. pp. 155-165 (Developments in Mathematics).
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Birmajer, D, Gil, JB, McNamara, PRW & Weiner, MD 2019, Enumeration of colored dyck paths via partial bell polynomials. in Developments in Mathematics. Developments in Mathematics, vol. 58, Springer New York LLC, pp. 155-165. https://doi.org/10.1007/978-3-030-11102-1_8

Enumeration of colored dyck paths via partial bell polynomials. / Birmajer, Daniel; Gil, Juan B.; McNamara, Peter R.W.; Weiner, Michael D.

Developments in Mathematics. Springer New York LLC, 2019. p. 155-165 (Developments in Mathematics; Vol. 58).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Birmajer D, Gil JB, McNamara PRW, Weiner MD. Enumeration of colored dyck paths via partial bell polynomials. In Developments in Mathematics. Springer New York LLC. 2019. p. 155-165. (Developments in Mathematics). https://doi.org/10.1007/978-3-030-11102-1_8