... invalid in the other.)This lemma, together with the discussion preceding it, completes the proof of formula(18). We leave to the reader the task of showing that this formula reduces to formula(4) ... t).In other words, the statistics for labelled paths inside the triangle with vertices (0, 0), (0,n),and (mn, n) give a combinatorial interpretation for the Hilbert series of the generalized diagonal ... diagram of λ. For example, the transpose of the partition in Figure 1 isλ=(7, 6, 5, 5, 3, 2, 2, 1).Definition 2. Let λ be a partition of N.Letc be one of the N cells in the diagram of λ.the...