Problem 2. Given an integer Y, the problem of Integer Partitioning aims to provide all possible ways of writing Y, as a sum of positive integers.
This problem can be mapped into the problem of Integer Partitioning as defined below.
The Gibbs partition is a family of measures on integer partition, and several prior processes, such as the Dirichlet process, naturally appear in connection with infinite exchangeable Gibbs partitions.
They play a prominent role in studying integer partition congruences.
This notation for an integer partition is referred to as the frequency representation of the integer partition.
Every partition of a set S induces an integer partition of | S|.
The true selfing line partition in our example induces an integer partition of the sample size 100.
I denote the set of all possible integer partitions of the Y pairs of populations as a(Y) and the length of that set as | a(Y)|, and I hereinafter refer to these integer partitions as "unordered" divergence models or partitions.
The first simply gives all possible unordered partitions of Y elements equal probability (14) p (t ) = 1 | a (Y ) |, i.e., a discrete uniform prior over all the integer partitions of Y (unordered divergence models).
This integer partition can be written as 123456, - meaning that there are 40 sets of size 1, 134 sets of size 2, and so on, in the original set partition.
So at first, we have to find the expected number of partitions, i.e., the expected number of runs of H. Kessler and Livingston [ 16] showed that to get an integer partition of an integer Y, expected number of required parts is: 3 Y 2 π × (log Y + 2 γ - 2 log π 6 ), where γ is the famous Euler's constant.
