Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [integer-partitions]

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

5
votes
1answer
713 views

identity proof for partitions of natural numbers

Definition: A tuple $\lambda = (\lambda_1, \cdots, \lambda_k)$ of Natural Numbers is called a numeric partition of n if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and $\lambda_1 + \cdots + \...
2
votes
1answer
690 views

How to find the coefficient of a term in this expression

How to determine the coefficient of z3q100 in I stumbled upon this problem while trying to solve this type of partition problem: Find the number of integer solutions to x + y + z = 100 such that 3 &...
5
votes
1answer
3k views

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 ...
6
votes
1answer
181 views

Why is there a derivative in this formula?

This is a very simple question. Why is Rademacher's formula presented with d/dx in it? Why not just "do" the derivative? Then replace x with n? Is it so there is only one transcendental function ...
25
votes
1answer
855 views

Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — and,...
0
votes
3answers
554 views

Partition an integer $n$ by limitation on size of the partition

According to my previous question, is there any idea about how I can count those decompositions with exactly $i$ members? for example there are $\lfloor \frac{n}{2} \rfloor$ for decompositions of $n$ ...
1
vote
3answers
305 views

Decomposition by subtraction

In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions: \begin{gather*} 13\\ 3,10\\ 4,9\\ 5,8\\ 6,7\\ 3,3,7\\ 3,4,6\\ 3,5,5\\...
1
vote
1answer
311 views

Seeking some details about what is denoted by the partition function $P(n,k)$

Quoting from Wolfram MathWorld, "$P(n,k)$ denotes the number of ways of writing $n$ as a sum of exactly $k$ terms or, equivalently, the number of partitions into parts of which the largest is exactly $...
5
votes
3answers
793 views

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 $\bar p_k(n)$ ...
9
votes
1answer
270 views

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 ...
4
votes
1answer
134 views

Validity of a q-series theorem

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: $$\...
5
votes
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 ...
3
votes
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 ...
2
votes
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 ...
1
vote
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 ...
8
votes
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 ...
23
votes
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 ...
11
votes
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 ...
27
votes
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 ...
2
votes
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 $$A(r,n)=\sum_{\nu=0}^{n-1}(-1)^{\nu}\binom{n}{\nu}(n-\nu)^{r}$$...
2
votes
1answer
461 views

Matrix representation of a partition

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?