We introduce a polynomialC̃μ[Z;q, t], depending on a set of variablesZ=z1,z2,..., a partition μ, and two extra parametersq, t. The definition ofC̃μinvolves a pair of statistics (maj(σ, μ), inv(σ, μ)) on words σ of positive integers, and the coefficients of theziare manifestly in\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{N}}[q,t]\end{equation*}\end{document}. We conjecture thatC̃μ[Z;q, t] is none other than the modified Macdonald polynomialH̃μ[Z;q, t]. We further introduce a general family of polynomialsFT[Z;q, S], whereTis an arbitrary set of squares in the first quadrant of thexyplane, andSis an arbitrary subset ofT. The coefficients of theFT[Z;q, S] are in\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{N}}[q]\end{equation*}\end{document}, andC̃μ[Z;q, t] is a sum of certainFT[Z;q, S] times nonnegative powers oft. We proveFT[Z;q, S] is symmetric in theziand satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial inFT[Z;q, S] can be expressed recursively.maplecalculations indicate theFT[Z;q, S] are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the setTis a partition with at most three columns.