Three New Refined Arnold Families

Sen Peng Eu, Louis Kao

Research output: Contribution to journalArticlepeer-review


The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array (vn,k) of integers, 1≤ |k| ≤ n, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects (Xn,k) is an Arnold family if Xn,k is counted by vn,k. A polynomial refinement Vn,k (t) of vn,k, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth’s flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.

Original languageEnglish
Article numberP4.19
JournalElectronic Journal of Combinatorics
Issue number4
Publication statusPublished - 2023

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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