To solve this, we need to first convert all the mixed numbers to improper fractions:

2 5/9 = 18/9 + 5/9 = 23/9
1 20/21 = 21/21 + 20/21 = 41/21
1 8/49 = 49/49 + 8/49 = 57/49

Now we can perform the operations:

(23/9) - (41/21) ÷ (57/49) + (8/9) : 6
(23/9) - (41/21) ÷ (57/49) + (8/9) × (1/6)
(23/9) - (41/21) ÷ (57/49) + (4/9)
(23/9) - (41/21) ÷ (57/49) + (4/9) × (7/7)
(23/9) - (41/21) ÷ (57/49) + (28/63)
(23/9) - (41/21) ÷ (57/49) + (28/63) × (9/9)
(23/9) - (41/21) ÷ (57/49) + (252/441)
(23/9) - (41/21) ÷ (57/49) + (252/441) = (23/9) - (41/21) ÷ (57/49) + (252/441)
= (23/9) - (82/63) + (252/441)
= (23/9) - (82/63) + (4/7)
= (35/63) + (4/7)
= (245 + 36) / 441
= 281/441

Therefore, the final result is 281/441.