Three New Refined Arnold Families

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 (v_n,k) of integers, 1≤ |k| ≤ n, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects (X_n,k) is an Arnold family if X_n,k is counted by v_n,k. A polynomial refinement V_n,k (t) of v_n,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.