TY - JOUR

T1 - Springer Numbers and Arnold Families Revisited

AU - Eu, Sen Peng

AU - Fu, Tung Shan

N1 - Funding Information:
The authors thank the referees for providing helpful suggestions. The authors were supported in part by the Ministry of Science and Technology (MOST), Taiwan under grants 110-2115-M-003-011-MY3 (S.-P. Eu), and 109-2115-M-153-004-MY2 (T.-S. Fu).
Publisher Copyright:
© 2023, Institute for Mathematical Sciences (IMS), Stony Brook University, NY.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - Alternating permutation

KW - Euler number

KW - Signed permutation

KW - Springer number

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U2 - 10.1007/s40598-023-00230-9

DO - 10.1007/s40598-023-00230-9

M3 - Article

AN - SCOPUS:85153758920

SN - 2199-6792

JO - Arnold Mathematical Journal

JF - Arnold Mathematical Journal

ER -