# Questions tagged [integer-partitions]

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

226 questions
528 views

### Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
133 views

### Closed form for $\sum_{n=1}^\infty \frac{1}{P(n)}$, where $P(n)$ is the partition function.

Is there a closed form for the following infinite series? $$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$ where $P(n)$ is the partition function.
567 views

138 views

### 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....
153 views

199 views

### A general Combinatorics problem (Coefficients of the q factorial)

I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...
126 views

51 views

### An upper bound for integer partitions with unique summands

Let $p_\neq (n)$ be the number of all partitions of $n$ such that all summands are distinct (for example $p_\neq (6)=4$). How do we show that $p_\neq (n) \leq e^{2\sqrt n}$?
49 views

### Degree of generating polynomial associated with two partitions

Let $\lambda=(\lambda_1,\ldots,\lambda_k)$ and $\mu=(\mu_1,\ldots,\mu_r)$ be two strictly increasing sequences of non-negative integers ($r$ and $k$ may be $0$, in which case the sequence is empty, ...
156 views

### Is there a generating function for this sequence?

The sequence is: 1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,... It is related to partitions of $n$. It is a ...
42 views

### Partitions reference

In this paper, the author mentions before Theorem 23 that the result was proven by Elmo Moore in 1968; the associated reference says "unpublished result". I would be grateful to anyone who knows a ...
139 views

### Counting Semistandard Young Tableaux For Triangular Shapes?

If $k \leq n$ I denote the Young diagram with shape $(n,n-1,n-2,\ldots,1)$ by $\lambda^{n,n-1,\ldots,1}$. I write $f^{\lambda_n^{n,n-1,\ldots,1}}$ to count the number of semistandard Young tableaux ...
74 views

### Equal and unequal partition?

Can someone please tell me what equal and unequal partition is? And if the question, I'm just putting an example here, is asking you to find at least 3 equal and unequal partitions of 2020 into 4 ...
58 views

### Is there accepted notation for the set of ways of partitioning a natural number $a$ into $b$ parts?

Recall the following: Multinomial Theorem. For all finite sets $X$, we have: $$\left(\sum_{x \in X} x\right)^n = \sum_{a}[a](X)^a$$ where $a$ ranges over the set of partitions of $n$ into ...
196 views

### $4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$

To prove the identity $$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$$ I replaced $m-n$ by $k$ in LHS ...
79 views

### Schengen Visa rules

If a tourist wants to visit europe with a schengen visa, the following rules apply: maximum contiguous stay is 90 days (in the following treated as 3 months) in any 180 day interval the sum of days ...
89 views

### Partition set N into roughly-equal groups of size approximately k. How can I determine the “range” within which k can vary?

I'm a long way from uni, so please forgive me if I'm not presenting this question as clearly or concisely as I could. I've searched for similar questions, but they seemed to focus on "how many ways ...
84 views

### The number of partitions of $n$…Subbarao

(Subbarao) The number of partitions of $n$ in which each part appears two, three, or five times equals the number of partitions of $n$ into parts congruent to $2, 3, 6, 9,$ or $10$ modulo $12$. ...
113 views

57 views

### What are the combinatorial numbers appearing in these repeated derivatives?

Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$. I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$. Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
The partition function $p(n)$ measures the number of partitions of $n$, or the number of ways in which natural numbers can be summed to produce $n$, without regard to order. For example, the ...
### Is every integer $\geq5$ the sum of two primes and a power of a prime?
Is every integer $\geq5$ the sum of two primes and a power of a prime (where $1$ is included in the prime powers)? I don't really expect someone to prove this here, but I wonder if the question has ...