Explicit formula for the refined Eulerian generating function

Determine an explicit formula for the exponential generating function A(x, y, z₁, z₂; t) that enumerates permutations by the numbers of double descents, double ascents, type 1 peaks, and type 2 peaks, beyond the Riccati equation established in the paper.

Background

Section 3.1 introduces the refined Eulerian polynomials Aₙ(x, y, z₁, z₂), where x, y, z₁, and z₂ mark double descents, double ascents, type 1 peaks, and type 2 peaks, respectively. The associated exponential generating function A(x, y, z₁, z₂; t) is related to refined statistics on weakly increasing trees and generalizes a known Carlitz–Scoville generating-function formula when z₁ = z₂.

The authors derive a Riccati differential equation for the fully refined generating function but do not obtain a closed explicit expression. They solve only the special case x = y = z₂ = 1, leaving the general explicit-formula problem unresolved.

References

Unfortunately, we are unable to find any explicit formula for Apx, y, z1, z2; tq. Instead, we can prove that Apx, y, z1, z2; tq satisfies a Riccati equation.

— Bijections in weakly increasing trees via binary trees  (2502.09161 - Li et al., 13 Feb 2025) in Section 3.1, p. 16