A family of Bell transformations

Daniel Birmajer, Juan B. Gil, Michael D. Weiner

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full range of applications remains unexplored, in this paper we show a glimpse of the versatility of Bell transformations by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations.

Original languageEnglish (US)
Pages (from-to)38-54
Number of pages17
JournalDiscrete Mathematics
Issue number1
StatePublished - Jan 2019

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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