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This is a question for applications of partitions in science or technology. I know partitions is an interesting field in combinatorics and in modern algebra because it can be related with symmetric polynomials, for example. My question is that if this has some application in other areas. I am thinking for example in computer science, cryptography, or even physics/chemistry.

I would be very glad, from the experts on the field, if someone can provide references if such applications exist.

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  • $\begingroup$ Partitions of whole numbers and partitions of sets are somewhat different subjects in mathematics, and you've not indicated which of these is your target. Narrowing the Question will likely help to expedite the kinds of answers about applications you are looking for. $\endgroup$ – hardmath Mar 11 '16 at 18:29
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    $\begingroup$ Considering the question mentions that they are related to symmetric polynomials, I think it's reasonable to assume that partitions of whole numbers are up for consideration here :) $\endgroup$ – Eric Stucky Mar 11 '16 at 19:20
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There are many such applications. A straightforward one is that partitions can be used in statistical mechanics to count available states to many-particle bosonic/fermionic systems and in the calculation of their "partition" functions.

A particularly important application is that partitions label irreducible representations of important groups like the permutation group $S_n$ and the unitary group $U(n)$, which themselves have applications in molecular chemistry, crystalography and quantum mechanics.

As you have mentioned, they also label basis elements for the vector space of homogeneous symmetric polynomials, and these are important for computing some many-variable integrals, for representing wave functions of many-body quantum systems and in the statistical theory of random matrices that is used to model complex networks, disordered media and chaotic quantum systems, for instance.

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Partitions of integers appear in genetics. Google the terms "Ewens sampling formula" and "Chinese restaurant process".

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