Non-recursive formulas for Graham-positive expansions

Develop non-recursive combinatorial formulas that isolate the manifestly Graham-positive output of the recursion for long-range divided differences associated with noncrossing alternating forests.

Background

The paper establishes Graham-positive expansions for expressions obtained by applying composites of long-range divided differences to double Schubert polynomials. These expansions are produced by a recursive straightening algorithm involving noncrossing alternating forests and related divided-difference identities.

Although the algorithm proves positivity, its output is not presented through direct non-recursive combinatorial formulas. The authors explicitly leave the task of isolating such formulas unresolved, making this a problem about obtaining a more transparent and potentially more efficient description of the positive coefficients.