Shade in partitions

Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing)
partitions of \( n \).