a.py

import math


m = [
    [1, 2, 4, 7, 11, 16],
    [3, 5, 8, 12, 17],
    [6, 9, 13, 18],
    [10, 14, 19],
    [15, 20],
    [21],
]


def inv(k):
    x = int((math.sqrt(8 * k - 7) + 1) / 2)
    y = k - x * (x - 1) / 2
    x -= y - 1
    return x, y


for row in m:
    print "\t".join(str(inv(k)) for k in row)

out.txt

(1, 1)	(2, 1)	(3, 1)	(4, 1)	(5, 1)	(6, 1)
(1, 2)	(2, 2)	(3, 2)	(4, 2)	(5, 2)
(1, 3)	(2, 3)	(3, 3)	(4, 3)
(1, 4)	(2, 4)	(3, 4)
(1, 5)	(2, 5)
(1, 6)