We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little $q$-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little $q$-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Young diagram, have a product formula. We also explore various combinatorial properties of lecture hall tableaux.

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演讲厅画面

我们介绍演讲厅画面，它是满足特定条件的偏斜杨氏图的填充物。演讲厅画面概括了演讲厅隔断和反演讲厅构图，并且还包含反向半标准 Young 画面作为极限情况。我们表明，多元小 $q$-Jacobi 多项式的舒尔展开中的系数是演讲厅画面的生成函数。使用 Selberg 型积分，我们证明多元小 $q$-Jacobi 多项式的矩等于杨氏图的演讲厅画面的生成函数，具有乘积公式。我们还探索了演讲厅画面的各种组合特性。