Springer Numbers and Arnold Families Revisited

Sen Peng Eu, Tung Shan Fu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


For the calculation of Springer numbers (of root systems) of type Bn and Dn , Arnold introduced a signed analogue of alternating permutations, called βn -snakes, and derived recurrence relations for enumerating the βn -snakes starting with k. The results are presented in the form of double triangular arrays (vn,k) of integers, 1 ≤ | k| ≤ n . An Arnold family is a sequence of sets of such objects as βn -snakes that are counted by (vn,k) . As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of tan x and sec x , established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.

Original languageEnglish
JournalArnold Mathematical Journal
Publication statusAccepted/In press - 2023


  • Alternating permutation
  • Euler number
  • Signed permutation
  • Springer number

ASJC Scopus subject areas

  • Mathematics(all)


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