Inclusion-exclusion on Schubert polynomials (Q2136400)

\textit{N. Fan} and \textit{P. Guo} [Sci. China, Math. 65, No. 6, 1319--1330 (2022; Zbl 1490.05272)] proved a combinatorial formula for the Schubert polynomial of any permutation which avoids the patterns 1423 and 1432. The authors use this formula to prove the nonnegativity of an inclusion-exclusion-inspired formula for these Schubert polynomials, in which the coefficients are given by combinatorial expressions in the Rothe diagrams of the permutations. \textit{Y. Gao} [Eur. J. Comb. 94, Article ID 103291, 12 p. (2021; Zbl 1462.05358)] conjectured that the principal specialization \({\mathfrak S}_w(1,\dots,1)\) of the Schubert polynomial is a sum of positive constants \(c_{\mathrm {perm}(v)}\) over the permutations corresponding to all subwords \(v\) of \(w\). The inclusion-exclusion formula specializes in the Möbius inversion of the conjectured formula, and thus proves the conjecture for permutations avoiding 1432 and 1423. There are generalizations of the main lemmas which lead to conjectures and a possible framework for similar results for arbitrary permutations.