On rational Dyck paths and the enumeration of factor-free Dyck words

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

Original languageEnglish (US)
Pages (from-to)36-43
Number of pages8
JournalDiscrete Applied Mathematics
Volume244
DOIs
StatePublished - Jul 31 2018

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Dyck Paths
Enumeration
Statistics
Polynomials
Bijection
Bell Polynomials
Ascent
Reducibility
Statistic
Partial
Path
Subset
Alternatives

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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On rational Dyck paths and the enumeration of factor-free Dyck words. / Birmajer, Daniel; Gil, Juan Bautista; Weiner, Michael David.

In: Discrete Applied Mathematics, Vol. 244, 31.07.2018, p. 36-43.

Research output: Contribution to journalArticle

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AB - Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alternative formulas for various enumerative sequences that appear in the context of rational Dyck paths.

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