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I know that the question of a closed form for the number of partitions of $n$, often written $p(n)$, is an open one (perhaps answered by the paper referred to in this question's answer, although I'm not a mathematician so have no understanding of that stuff really).

However, my question is: is there a 'closed form'/formulaic way, call it $\lambda(n)$, to write down the content or form of the partitions, e.g. for $n=4$, while $p(n)=5$, the content of the partitions is (using the exponent to show multiplicity a la cycle type notation):

$\lambda(4) = \{(1^4), (1^2,2), (1,3), (2^2), (4)\}$

I am particularly interested (because of other questions of mine - here and here), in being able to then exclude digits (which account for cycle sizes), i.e. the non-unity partitions excluding all fixed point 1-cycles.

I know that there exist algorithmic ways, e.g. Jerome Kelleher's accel_asc() (pointed to me by this question), but these are not as helpful in writing mathematical expressions... or if they are, I am not sufficiently skilled in doing so...

... So perhaps my question then becomes: how to convert the algorithmic way into a more closed form/formulaic algebraic expression?

$\mathbf{EDIT:}$ As pointed out by the comment below - this would necessarily just give $p(n)$, which does not exist as of yet, apparently! I'll leave the question open just in case someone can help me express / understand the algorithmic method better mathematically?

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    $\begingroup$ A closed/formulaic expression to calculate $\lambda(n)$ will immediately give you a closed expression for $p(n)$ (just calculate $\lambda(n)$ and then count its elements), and you already observed that $p(n)$ has no closed form. $\endgroup$
    – jjagmath
    Commented May 20 at 12:06
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    $\begingroup$ $p(4)=5$, not $4$. $\endgroup$
    – Gerry Myerson
    Commented May 20 at 12:19
  • $\begingroup$ Oops sorry yes, I forgot $(4)$ itself. Corrected! @jjagmath good point... :/ ... But I'll leave the question open just in case someone can help me understand a better way of mathematically expressing the algorithmic way? $\endgroup$
    – julianiacoponi
    Commented May 20 at 12:24

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Integer partitions are generated and enumerated by the exponential Bell polynomials.

If you want to generate only the parts structure of the partitions, you could remove the coefficient in the formula of the exponential Bell polynomials.

But the the formula of the exponential Bell polynomials needs to generate the partions: see the constraints for the running index of the sum sign.

The better way is to use a software algorithm to generate the partitions or the parts structure of the partitions.

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  • $\begingroup$ Thanks @IV_ for your answer! Much appreciated. Could you clarify what you mean by the "constraints for the running index of the sum sign"? $\endgroup$
    – julianiacoponi
    Commented May 24 at 9:40
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    $\begingroup$ @julianiacoponi See the formula in the Wikipedia article: $ j_{1}+j_{2}+\cdots +j_{n-k+1}=k, j_{1}+2j_{2}+3j_{3}+\cdots +(n-k+1)j_{n-k+1}=n$. That means the partions itself (means number and length of the parts for each partition) are needed before the formula can be applied. $\endgroup$
    – IV_
    Commented May 24 at 10:03

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