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Note: These are not the actual ratios, this is just an example.

Luck is player stat that reduces a players chance of finding items of normal quality. Reducing the ratio of normal quality drops effectively increases the chance that higher item qualities will drop.
In a party the Luck of the player who deals the killing blow is what determines drops. Same is applicable to opening chests - the loot will be generated for party members in a ~50m range according to the Luck of the player who opened it. For example:

Item Quality Weighting Percent
Common 1300 90.9%
Enhanced 100 7%
Rare 25 1.7%
Legendary 5 0.35%
Unique 0.8 0.06%
Mythic 0.1 0.007%

Luck subtracts itself from the Common quality pool while retaining all other quality weightings, effectively increasing chance of finding items of higher quality.
Let's say, the player has 1300 Luck, which is enough to "remove" Common from drop pool. The result will be:

Item Quality Weighting Percent
Common 0 0%
Enhanced 100 76.4%
Rare 25 19%
Legendary 5 3.8%
Unique 0.8 0.61%
Mythic 0.1 0.076%

As you can see, removing Common from drop pool has led to an x10.5 increase in drop rates of other items.
Please note that Depending on the enemy grade (25% for Champions, 50% for bosses), there is a chance to simply bypass Commons or Commons+Enhanced respectively. This is not just assuming ~1300 Luck (ie 1/4 or 1/2 of the time you'll get max Luck drops), but rather applying ~1300 Luck to a new pool which just lacks Commons or Commons and Enhanced. Therefore, in those cases you'd have different results if you had 0 Luck and if you had ~1300 from enriched rolls.[1]

Influence vs Bosses

Bosses and some unique named monsters may have two drop tables, one table with improved drop rates and is affected by Luck, and the other is predetermined with fixed drop chances for monster specific Unique and Named items, on which Luck has no effect whatsoever. Therefore, using the above example the non-predetermined drop table will appear:

Item Quality Weighting Percent
Common 0 0%
Enhanced 0 0%
Rare 25 80.9%
Legendary 5 16.2%
Unique 0.8 2.6%
Mythic 0.1 0.3%
Item Quality Weighting Percent
Common 0 0%
Enhanced 0 0%
Rare 0 0%
Legendary 5 84.7%
Unique 0.8 13.6%
Mythic 0.1 1.7%