I was considering poker hands in an attempt to generalise the game to any number of suits and cards per hand. Apart from flushes, straights and combinations thereof, the poker hands are (in descending order): Four of a kind (4,1,0,0,0,0,0,0,0,0,0,0,0) Full house (3,2,0,0,0,0,0,0,0,0,0,0,0) Three of a kind (3,1,1,0,0,0,0,0,0,0,0,0,0) Two pairs (2,2,1,0,0,0,0,0,0,0,0,0,0) Pair (2,1,1,1,0,0,0,0,0,0,0,0,0) High card (1,1,1,1,1,0,0,0,0,0,0,0,0) You should be able to determine the notation used based on context. Notice that each sequence majorises all lower sequences, in the way required for Muirhead's Inequality to hold. http://en.wikipedia.org/wiki/Muirhead's_inequality Now, it is not the case that if one sequence majorises the other, then it is a more likely poker hand. For example, consider a variant with 13 cards per hand. It is obvious that (1,1,1,1,1,1,1,1,1,1,1,1,1) is less likely than (2,1,1,1,1,1,1,1,1,1,1,1,0), yet the latter majorises the former. However, if we consider the limit of poker games where the number of cards per suit increases to infinity, then A is rarer than B if sequence A majorises sequence B. What about sequences where neither one majorises the other? The minimal example is (3,3,0) compared with (4,1,1), which only occurs in six-cards-per-hand poker. Sincerely, Adam P. Goucher