Shade in partitions

Abstract

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 $n$.