Dyck-path-compatible wreath decomposition

Construct, for every positive integer k, a family of C_k permutations of Z_{2k+1}, each fixing 0, together with a bijection from these permutations to Dyck k-paths, such that the associated wreaths partition the k-subsets of Z_{2k+1} and the j-th step of the Dyck path corresponding to a permutation is a rise exactly when the permutation value at j belongs to {1,...,k}.

Background

For n = 2k + 1, every wreath has 2k + 1 members, and the number of wreaths required to partition the k-subsets of Z_{2k+1} equals the Catalan number C_k, which is also the number of Dyck paths of semilength k.

The authors propose strengthening the general wreath conjecture by requiring the wreath-generating permutations to be indexed by Dyck paths. The rise/fall condition encodes whether permutation values lie in the first or second half of the nonzero elements of Z_{2k+1}. The conjecture was verified computationally for k ≤ 4.