# Questions tagged [integer-partitions]

Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.

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### Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's proof....
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### Asymptotic behavior of unique integer partitions

Okay, this is one of those questions that I'm sure has a very simple answer I'm missing, and I'd appreciate any push in the right direction. Consider a very large integer $N$. Stealing an example ...
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### Partitions of a prime power into powers of the same prime

Fix a prime $p$, and $k$ a natural number. The question is then: How many partitions of $p^k$ are there into powers of $p$? So, for instance, if $p = 2$ and $k = 2$, there are 4, namely (4), (2, 2), ...
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### For every partition $\pi$ of a fixed integer $n$, $\sum{F(\pi)}=\sum{G(\pi)}$

I need to prove the following question. For every partition $\pi$ of a fixed integer $n$, define $F(\pi)$=number of occurrences of 1 as a summand, and $G(\pi)$=no. of distinct summands in the ...
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### Generating function for $r^\binom{n}{2}$

I'm trying to find a closed form of the generating function $$G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n$$ for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
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### Representations of an integer as the sum of other integers

Given a finite set $S$ of (distinct) integers $s_1, \dots, s_n$ and an integer $x$, I'm looking for all representations (where order is important) $$x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\})$$ ...
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### Creating generating functions for integer partitions

Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the ...
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### Is every positive integer the sum of 1 square number, 1 pentagonal number, and 1 hexagonal number?

I found this interesting conjecture, but maybe I'm not the first to state it. I have tested it for the first $10^4$ positive integers, but that is not a proof. Can anybody prove or disprove this ...
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### Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) (...
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### Average length of partitions

I'm wondering if there's a known asymptotic for an average number of terms for partitions of a given number? (I mean, given all the partitions of a given number, how many terms do they have on average?...
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### Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
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### Degree distribution of a graph of integer partitions

Consider a graph whose nodes are the integer partitions of $n$, connected by an edge if you can get from one partition to the other by ‘sliding one square of its Young diagram’ – that is, if one ...
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### Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
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### How many combinations of $3$ natural numbers are there that add up to $30$?

How many combinations of $3$ natural numbers are there that add up to $30$? The answer is $75$ but I need the approach. Although I know that we can use $_{(n-1)}C_{(r-1)}$ i.e. $_{29}C_2 = 406$ but ...
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### Partition Bijection

I'm not sure what I'm missing. I think I'm thinking too hard about finding this bijection. Please help!
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### Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: 1) ...
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### Number of permutations with a given partition of cycle sizes

Part of my overly complicated attempt at the Google CodeJam GoroSort problem involved computing the number of permutations with a given partition of cycle sizes. Or equivalently, the probability of a ...
### Combinatorial interpretation of $n P_n = \sum_{j=1}^n \, \sigma(j) P_{n-j}$?
By logarithmic differentiation, one can deduce that $n P_n = \sum_{j=1}^n \, \sigma(j) P_{n-j}$ where $P_n$ is the $n$-th partition number, and $\sigma(j)$ the sum of the divisors of $j$. I see ...