A partition of a positive integer n (or a partition of weight n) is a non-decreasing sequence λ = (λ1, λ2, . . . , λk) of non-negative integers λi such that ∑k i=1 λi = n. The λi’s are the parts of the partition λ. Integer partitions are of particular interest in combinatorics, partly because many profound questions concerning integer partitions, solved and unsolved, are easily stated, but not ...

We introduce generous, even-matched, and greedy strategies as concepts for analyzing games. A two person prisoner's dilemma game is described by the four outcomes (C,D), (C,C), (D,C), and (D,D). In a generous strategy the proportion of (C,D) is larger than that of (D,C), i.e. the probability of facing a defecting agent is larger than the probability of defecting. An even-matched strategy has th...

A partition structure is a sequence of probability distributions for πn, a random partition of n, such that if πn is regarded as a random allocation of n unlabeled balls into some random number of unlabeled boxes, and given πn some x of the n balls are removed by uniform random deletion without replacement, the remaining random partition of n − x is distributed like πn−x, for all 1 ≤ x ≤ n. We ...

So there 5 partitions for 4. Assign each partition with a probability, we then get a probability measure on them. Different ways of assigning probabilities: Uniform measure Plancherel measure Jack measure Restricted uniform measure Restricted Jack measure. We will study the properties of r.v.’s .k1; k2; : : :/, a random partition of n. Example 2. As n!1, under the Plancherel measure, k1 2 p n n...

We characterize all numbers n and S with the following property: Every instance of the partition problem that consists of n positive integers with sum S possesses a solution, that is, a partition into two subsets with equal sum. c © 2006 Published by Elsevier Ltd

upon specializing q = qτ := e 2πiτ , τ ∈ H the upper complex half-plane, where η(τ) := q1/24 ∏ n≥1(1 − qn) is Dedekind’s η-function, a weight 1/2 modular form. More recently, Bringmann and Ono [8] studied the generating function for partition ranks, where the rank of a partition, after Dyson, is defined to be the largest part of the partition minus the number of parts. For example, the rank of ...

So there 5 partitions for 4. Assign each partition with a probability, we then get a probability measure on them. Different ways of assigning probabilities: Uniform measure Plancherel measure Jack measure Restricted uniform measure Restricted Jack measure. We will study the properties of r.v.’s (k1, k2, . . .), a random partition of n. Example 2. As n→∞, under the Plancherel measure, k1 − 2 √ n...

For any partition of {1, 2, . . . , n} we define its increments Xi, 1 ≤ i ≤ n by Xi = 1 if i is the smallest element in the partition block that contains it, Xi = 0 otherwise. We prove that for partially exchangeable random partitions (where the probability of a partition depends only on its block sizes in order of appearance), the law of the increments uniquely determines the law of the partit...