Combinatorial interpretation of bi-γ-coefficients for general 1^k-Eulerian polynomials

Provide a combinatorial interpretation of the bi-γ-coefficients of the 1^k-Eulerian polynomials A_n^{(k)}(x) for arbitrary k.

Background

The 1^k-Eulerian polynomials A_n^{(k)}(x) admit a symmetric decomposition whose two components have combinatorial interpretations in terms of k-Stirling permutations. Prior work established bi-γ-positivity for these polynomials and gave a combinatorial interpretation of the bi-γ-coefficients in the special case k=2.

The unresolved case concerns arbitrary k. The paper addresses this problem by introducing increasing pruned even k-ary forests, constructing bijections with k-Stirling permutations, and using generalized Foata–Strehl actions and further forest transformations to interpret the bi-γ-coefficients combinatorially.