Before revision:  Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), } and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X_0(\mathcal{K}} ) can all be efficiently computed once the elder-rule-staircode is given . Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration .
After revision:  Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), } and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X_0(\mathcal{K}} ) can all be efficiently computed once the elder-rule-staircode is given .