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 1k-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.

References

The combinatorial interpretation for the bi-γ-coefficients of A(k)n (x) for general k still remains open.

— Combinatorics on bi-$γ$-positivity of $1/k$-Eulerian polynomials  (2501.12055 - Yan et al., 21 Jan 2025) in Section 1 (Introduction), following Proposition 1.2, p. 2

Is there a Foata--Strehl-type or valley-hopping action on each fixed cycle-type stratum whose orbit polynomials are $tk(1+t){2j+f-2k}$ and whose primitive objects are enumerated by eq:A-explicit?

eq:A-explicit:

Aj,k,r=∑s=0jD2j,s ⁣∑a,b,c≥0a+c=k−sa+b+2c=rb+c≤s4a+c2b(2j−2sa)(sb,c,s−b−c).A_{j,k,r} =\sum_{s=0}^{j}D_{2j,s} \!\sum_{\substack{a,b,c\ge0\\ a+c=k-s\\ a+b+2c=r\\ b+c\le s}} 4^{a+c}2^b \binom{2j-2s}{a} \binom{s}{b,c,s-b-c}.

— A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions  (2609.04922 - Zeng, 4 Sep 2026) in Section Further questions, item 2