Binomial Coefficients and Littlewood--Richardson Coefficients for Interpolation Polynomials

Inhomogeneous versions of Jack and Macdonald polynomials, called interpolation polynomials, have been introduced by Knop--Sahi (type A) and Okounkov (type $BC$). In this paper, we study binomial coefficients and Littlewood--Richardson (LR) coefficients for these interpolation polynomials. We extend to type $BC$ the weighted sum formula for binomial coefficients due to the second author in type A, and obtain a new weighted sum formula for LR coefficients for both types A and $BC$. We prove that binomial coefficients are positive and monotone using the weighted sum formula and the combinatorial formulas due to Okounkov. As an application, we show that the containment partial order can be characterized in terms of Schur positivity or Jack positivity. This result is in parallel with the work of Cuttler--Greene--Skandera, Sra and Khare--Tao, which characterize two other partial orders, majorization and weak majorization, in terms of evaluation positivity of Schur functions.