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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$\mathfrak{sl}(2)$ decomposition for classical nilpotent orbits

Consider the nilpotent orbit, $\mathcal{O}_X$, of a semi-simple Lie algebra, $\mathfrak{g}$, represented by a nilpotent element $X\in \mathfrak{g}$. We can then always find a triplet $\{H,X,Y\}$ ...
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What is the closed form solution to the sum of inverse products of parts in all compositions of n?

My question is exactly as in the title: What is the generating function or closed form solution to the sum of inverse products of parts in all compositions of $n$? This question was inspired by just ...
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Number of ways of distributing $n$ elements

Suppose we have $n$ elements that we want to distribute to people forming a queue, and we want to know the number of different ways we could make the distribution independently of the number of ...
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What is the generating function for the number of partitions of an integer in which each part is used an even number of times?

What is the generating function for the number of partitions of an integer in which each part is used an even number of times? I'm trying to prove the number of partitions of an integer in which each ...
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Proving coefficient of $x^n$ in an infinite product of $(1+x^k)$ is the number of partitions of $n$ where each summand is distinct [closed]

Consider the number $q_n$ of partitions of n with all summands distinct. Prove that $q_n$ is the coefficient of $x^n$ in the infinite product $$\prod_{k=1}^\infty (1+x^k)$$ I solved a similar problem ...
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A counting problem with Young Diagrams

Let $n$ and $d$ be natural numbers. How many Young diagrams of size $n$ are there such that each diagram has less than or equal to $d$-rows ?
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Prove combinatorially the recurrence $p_n(k) = p_n(k−n) + p_{n−1}(k−1)$ for all $0<n≤k$.

Recall that $p_n(k)$ counts the number of partitions of $k$ into exactly $n$ positive parts (or, alternatively, into any number of parts the largest of which has size $n$).
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"On the Probability of Sequences in the Genoese Lottery" by Euler. How did he do it?

This a problem solved by Leonard Euler. Translated English version is available in Euler archives.[E338]On the Probability of Sequences in the Genoese Lottery Euler. I have difficulty in understanding ...
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Identity with integer partitions

I have to prove that $p(n)=p(n-1)+\displaystyle\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}p_{k}(n-k)$ and I am quite stuck on it... My first intuition was that, as $p(n)-p(n-1)$ is the number of partitions ...
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Partitions of $n$ where every element of the partition is different from 1 is $p(n)-p(n-1)$

I am trying to prove that $p(n|$ every element in the partition is different of $1)=p(n)-p(n-1)$, and I am quite lost... I have tried first giving a biyection between some sets, trying to draw an ...
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Partition function of multi-particle canonical ensemble

According to the orthogonality of function basis, Why can't the partition function be written directly as the following form \begin{align} \begin{split} Z & = \frac{1}{N!} \sum_{p_{1},p_{2}, \...
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Let g_n equal the number of lists of any length taken from {1,3,4} having elements that sum to n.

For example, g_3 = 2 because the lists are (3) abd (1,1,1). Also g_4 = 4 because the lsits are (4), (3,1), (1,3), and (1,1,1,1). Define g_0 = 1. (a) Find g_1, g_2, and g_5 by complete enumeration. (b) ...
For what values of $n≥1$ do there exist a number $m$ that can be written in the form $$a_1 + \cdots+ a_n$$ with $$a_1 \in \{1\}, a_2 \in \{1,2\},\cdots , a_n \in \{1,\ldots,n \}$$ in $(n-1)!$ or ...