文档介绍：: .
Dis represent an (n, ) dice by a partition of into n parts.
Definition 1. The n-tuple = (i1,i2,...,in) consisting of n strictly positive integers ordered non-decreasingly and
with collective sum equal to , is called a partition of into n parts. We will denote this type of partition by an (n, )
partition.
Note that in partition theory the parts are usually ordered non-increasingly. In the sequel, we shall often identify an
(n, ) dice Ai with its unique associated partition i of into n parts.
Once the players have chosen their own dice (note that they might have selected the same one), the (n, ) dice game
is played in one or more rounds. At the beginning of each round, both players place a bet of 1 say, then independently
roll their dice and compare the number of eyes on the bottom face: the dice that falls on the face with the highest num-
ber of eyes wins the round. The winner takes all and a new round can start. If the faces show the same number of eyes, the
E-mail addresses: Bart.******@ (B. De Schuymer), Hans.******@ (H. De Meyer), Bernard.******@
(B. De Baets).
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doi:. B. De Schuymer et al. / Discrete Applied Mathematics 154 (2006) 2565–2576
round ends in a draw: there is no winner, both players get their 1 back. Since the two players have as objective to
win the game, and since each round of the game proceeds under the same conditions (same strategy, same bet), they
want to choose from the collection a dice that maximizes their winning probability.
In order to compute the winning probability, let us regard dice Ai of player 1 and dice Aj of player 2 as independent
discrete random variables, uniformly distributed on the multiset consisting of the numbe