# 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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### Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)

Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ ...
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### Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition

Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...
Define the $q$-analog $(a;q)_n = \prod_{k=0}^n \left(1 - aq^k\right)$. I want to prove the identity $\frac{(q^2;q^2)_\infty}{(q;q)_\infty}=\frac{1}{(q;q^2)_\infty}$. I viewed the LHS this way: $$\... 2answers 403 views ### Natural set to express any natural number as sum of two in the set Any natural number can be expressed as the sum of three triangular numbers, or as four square numbers. The natural analog for expressing numbers as the sum of two others would apparently be the sum ... 2answers 901 views ### Graph coloring problem (possibly related to partitions) Given an undirected graph I'd like to color each node either black or red such that at most half of every node's neighbors have the same color as the node itself. As a first step, I'd like to show ... 1answer 2k views ### Upper bound on integer partitions of n into k parts Recent news piqued my interest in integer partitions again. I'm working my way back through an old text and I'm completely hung up on this problem: Recall that p_k(n) is the number of partitions ... 1answer 64 views ### Notation for “duplicating” partitions I'm using Macdonald's "Symmetric Functions and Hall Polynomials" as a reference and did not find what I was looking for -- apologies if I only missed it. As an example, let us consider the partition ... 1answer 4k views ### On problems of coins totaling to a given amount I don't know the proper terms to type into Google, so please pardon me for asking here first. While jingling around a few coins, I realized that one nice puzzle might be to figure out which n or so ... 0answers 538 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 ... 5answers 10k views ### Algorithm for generating integer partitions I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate all of them and then discard the ones that ... 7answers 20k views ### Making Change for a Dollar (and other number partitioning problems) I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ... 1answer 125 views ### Number of distributions leaving none of n cells empty The solution for the number of distributions leaving none of the n cells empty (with unlike cells and r unlike objects) is given by$$A(r,n)=\sum_{\nu=0}^{n-1}(-1)^{\nu}\binom{n}{\nu}(n-\nu)^{r}...
Is there a natural way to represent all the partitions of an integer set $\{1,2,3,...,n\}$ as a matrix in the similar way permutations can be mapped to group of matrices?