Configuration probability of partitioned objects

In summary, the conversation discusses the different ways to partition objects, whether they are distinct or identical. For distinct objects, there are K^N ways to partition them and for identical objects, there are (N+K-1)!/(N!*(K-1)!) ways. The probabilities for a specific configuration in the two cases are also mentioned. The conversation also mentions the coefficients of a polynomial that represents the number of ways to allocate identical objects to distinct partitions.
  • #1
rabbed
243
3
With N objects, if I arrange each without replacement into K distinct partitions in which different object orders should not be accounted for:

- For distinct objects I get a total number of Wtot = K^N ways to partition them, and a specific distribution with N1 objects in partition 1, N2 objects in partition 2 etc. can be accomplished in W = N!/(N1!*N2!*...*NK) ways.

- For identical objects I get a total number of Wtot = (N+K-1)!/(N!*(K-1)!) ways to partition them, and a specific distribution with N1 objects in partition 1, N2 objects in partition 2 etc. can be accomplished in W = 1 way.

So the probabilities of a specific configuration in the two cases should be:

Pdistinct = N!/(N1!*N2!*...*NK) / K^N
Pidentical = 1 / ( (N+K-1)!/(N!*(K-1)!) )

Is this correct?
 
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  • #2
That does look correct.
If you are interested in the number of ways to allocate ##n## identical objects to ##k## distinct partitions, consider the coefficients of the polynomial ##P_k(x) = \sum_{n=0}^{\infty} a_{k,n}x^n = (1-x)^{-k}##.
 
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Related to Configuration probability of partitioned objects

1. What is configuration probability of partitioned objects?

The configuration probability of partitioned objects refers to the likelihood that a certain arrangement or configuration of objects will occur when a larger set of objects is divided into smaller groups or partitions.

2. How is configuration probability calculated?

Configuration probability is typically calculated using mathematical formulas and statistical methods, such as combinatorics and probability theory. These calculations take into account the number of objects, the number of partitions, and any constraints or conditions on the arrangement of objects.

3. What factors can affect the configuration probability of partitioned objects?

The configuration probability of partitioned objects can be influenced by a variety of factors, including the number of objects, the number of partitions, the size and shape of the objects, and any constraints or rules governing the arrangement of objects.

4. How is configuration probability used in scientific research?

Configuration probability is commonly used in scientific research to analyze and understand patterns and relationships between objects. It can also be used to make predictions about the likelihood of certain configurations occurring in real-world scenarios.

5. Can configuration probability be applied to non-scientific fields?

Yes, configuration probability can be applied to various fields, such as economics, social sciences, and computer science. It can be used to analyze patterns and relationships in different systems and make predictions about their behavior.

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