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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1answer
114 views

Something basic in “l-adic properties of the partition function” paper

I am trying to understand the basic result in this paper: http://www.aimath.org/news/partition/folsom-kent-ono.pdf My problem is with the example at the end of page 2. I understand it's supposed to ...
9
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2answers
513 views

Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
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2answers
107 views

List all 3 part compositions of 5

I am looking at a past exam written by a student. There was a question I believed he got correct but received only 1/4. The marker wrote down "4 more compositions, order matters". This is the ...
4
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2answers
228 views

Trouble understanding proof that the unit interval cannot be partitioned in a certain way

From the book "Putnam and Beyond." The problem: Show that the interval [0, 1] cannot be partitioned into two disjoint sets A and B such that B = A + a for some real number a. Proof: Assume ...
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2answers
121 views

Generating sequences of numeric partitions

Definition: A tuple $\lambda = (\lambda_1, \ldots, \lambda_k)$ of natural numbers is called a numeric partition of $n$ if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and $\lambda_1 + \cdots + ...
5
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1answer
464 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
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1answer
506 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
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1answer
899 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
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1answer
157 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 ...
21
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1answer
565 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 $\left(\frac43\right)!$ — ...
0
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3answers
446 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 $n$ ...
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3answers
219 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\\ ...
1
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1answer
285 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 ...
4
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2answers
272 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 $\bar p_k(n)$ ...
7
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1answer
208 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 ...
3
votes
1answer
107 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
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2answers
319 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
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1answer
240 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
442 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
705 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
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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
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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 ...
13
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0answers
307 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
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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 ...
15
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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
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1answer
112 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
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3answers
605 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
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1answer
361 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?