Reflection-symmetric Dyck-path-compatible wreath decomposition

Construct, for every positive integer k, a Dyck-path-compatible wreath decomposition of the k-subsets of Z_{2k+1} satisfying the additional reflection symmetry that, for every Dyck k-path D and every j in Z_{2k+1}, the permutation assigned to D evaluated at j plus the permutation assigned to the reflected Dyck path D^(R) evaluated at -j equals 0.

Background

The stronger conjecture retains all requirements of the Dyck-path-compatible wreath decomposition and imposes a symmetry under reflection of Dyck paths across the vertical line x = k.

Specifically, the permutation associated with a Dyck path and the permutation associated with its reflected path must take opposite values at corresponding cyclic positions. The authors report that their computational examples satisfy this stronger conjecture for k ≤ 3.

References

Our search through $k \leq 4$ suggests that an even stronger statement may be true. To state it, for a Dyck path $D \in D{k}$, we will denote by $D{(R)}$ the Dyck path obtained by reflecting $D$ in the line $x=k$. Let $k$ be a positive integer. There exist a set $\Pi={\pi_1, \pi_2, \ldots, \pi_{#1{k}}$ of $C_k$ permutations with each permutation fixing 0 and a bijection $\varphi: \Pi \rightarrow D{k}$ such that

— Intervals in Dyck paths and the wreath conjecture  (2501.07277 - Petr et al., 13 Jan 2025) in Conjecture 2.3, Section 2 (the conjecture labelled \ref{conj:stronger})