to see this is to associate each word with a path across a rectangular grid with height and width , going from the bottom left corner to the top right corner. The path takes a step right for each 0 and a step up for each 1. An inversion switches the directions of a step (right+up becomes up+right and vice versa), hence the number of inversions equals the area under the path. Balls into bins Let be the number of ways of throwing indistinguishable balls into indistinguishable bins, where each bin can contain up to balls. The Gaussian binomial coefficient can be used to characterize . Indeed, where denotes the coefficient of in polynomial (see also Applications section below). Properties Reflection Like the ordinary binomial coefficients, the Gaussian binomial coefficients are center-symmetric, i.e., invariant under the reflection : In particular, Limit at q = 1 The evaluation of a Gaussian binomial coefficient at is i.e. the sum of the coefficients gives the corresponding binomial value. Analogs of Pascal's identity The analogs of Pascal's identity for the Gaussian binomial coefficients are: and When , these both give the usual binomial identity. We can see that as , both equations remain valid. The first Pascal analog allows computation of the Gaussian binomial coefficients recursively (with respect to m ) using the initial values and also shows that the Gaussian binomial coefficients are indeed polynomials (in q). The second Pascal analog follows from the first using the substitution and the invariance of the Gaussian binomial coefficients under the reflection . These identities have natural interpretations in terms of linear algebra. Recall that counts r-dimensional subspaces , and let be a projection with one-dimensional nullspace . The first identity comes from the bijection which takes to the subspace ; in case , the space is r-dimensional, and we must also keep track of the linear function whose graph is ; but in case , the space is (r−1)-dimensional, and we can reconstruct without any extra information. The second identity has a similar interpretation, taking to for an (m−1)-dimensional space , again splitting into two cases. Proofs of the analogs Both analogs can be proved by first noting that from the definition of , we have: As Equation () becomes: and substituting equation () gives the first analog. A similar process, using instead, gives the second analog. q-binomial theorem There is an analog of the