Alright, Carlton and Essendon have the most VFL/AFL premierships, with 16 each, but looking at who has won the most premierships is not a very interesting blog post (and doesn't fill our quota of at least one mathematical formula per post). Another question is which team has been the most "efficient" at winning premierships? To answer this, I've devised the following formula:
Premiership Efficiency Rating = (Premierships Won / Number of Seasons Played) - Average Probability Of Winning Premiership
The average probability of winning the premiership is calculated as the average of one divided by the number of teams over all the seasons a club has played. In 1897, the probability of winning was 0.125 (1/8); in 2011 it was 0.059 (1/17). Hence, the average probability is lower for newer clubs because a far higher proportion of their seasons were played in a larger league.
The "premiership efficiency rating" for each team is as follows:
Brisbane (Bears/Lions) 0.056
West Coast 0.056
Port Adelaide 0.004
North Melbourne -0.032
Gold Coast -0.059
Western Bulldogs -0.067
St. Kilda -0.078
Brisbane and West Coast come out on top, narrowly ahead of Essendon, assuming that the Brisbane Lions are counted as a continuation of the Brisbane Bears. If they're not, then the Brisbane Lions come out way ahead, with a PER of 0.138, as they have won 3 premierships since they began in 1997 in a league no smaller than 16 teams.
One could apply this formula to other competitions, and you would get more interesting results for the American professional sports associations, where there has been a wide variation in the degree of expansion over time. Indeed, I first had this idea in relation to the Davis Cup for tennis, in which the US and Australia won a lot of titles when fewer countries competed (and the format was different). Depending on how enthusiastic I feel, I might look at other leagues at a later date.