Before revision:  In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space . Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane.
After revision:  In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space . Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane.