Proof of a K-theoretic polynomial conjecture of Monical, Pechenik, and Searles

As part of a program to develop K-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas Āₐ = ∑b Qbᵃ(β)P̄b and Q̄ₐ = ∑b Mbᵃ(β)F̄b, where each of Āₐ, P̄ₐ, Q̄ₐ and F̄ₐ is a family of polynomials that forms a basis for Z[x₁,,xₙ][β] indexed by weak compositions $a,$ and Qbᵃ(β) and Mbᵃ(β) are monomials in β for each pair $(a,b)$ of weak compositions. The polynomials Āₐ are the Lascoux atoms, P̄ₐ are the kaons, Q̄ₐ are the quasiLascoux polynomials, and F̄ₐ are the glide polynomials; these are respectively the K-analogues of the Demazure atoms Aₐ, the fundamental particles Pₐ, the quasikey polynomials Qₐ, and the fundamental slide polynomials Fₐ. Monical, Pechenik, and Searles conjectured that for any fixed $a,$ ∑b Qbᵃ(-1), ∑b Mbᵃ(-1) ∈ \0,1\, where b ranges over all weak compositions. We prove this conjecture using a sign-reversing involution.