Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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7
votes
1answer
210 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
112 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: ...
4
votes
2answers
322 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 ...
6
votes
1answer
242 views

What is the number of ways to choose x groups from y items? (partitions with x cells of a multiset)

Where a group can consist of 1 or more items, groups don't have to be equally sized and items can be duplicates. Example - Choose 3 groups: Items: 1 2 2 3 Groups: (1) (2 2) (3) (1 2) (2) (3) (3 ...
3
votes
2answers
456 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 ...
1
vote
1answer
726 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
51 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
2k 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 ...
14
votes
0answers
318 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 ...
2
votes
4answers
4k 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 ...
17
votes
7answers
7k 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
113 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 ...
3
votes
3answers
622 views

“Converting” equivalence relations to partitions

There is a direct relationship between equivalence relations and partitions. Is there a way to simply use an equivalence relation's definition to get the matching partition? And what about the other ...
2
votes
1answer
367 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?