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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4
votes
1answer
500 views

Partitions of $\{1,\dots,n\}$ with no consecutive integers in each block is counted by $B(n-1)$?

I'm trying to understand why $B(n-1)$ also counts the number of partitions of $[n]$ where not two consecutive integers appear in the same block. Now the bell number $B(n-1)$ counts the number of ...
3
votes
1answer
685 views

Generating Function for Even Number of Odd Parts

How I would write the generating function for a partition of a positive integer n with an even number of odd parts? Any hints or suggestions will be greatly appreciated.
3
votes
1answer
223 views

number of additive partition

I have a question related with number of additive partition or method similar like this: $$p(5)=1+4=2+3=1+1+1+1+1=1+1+1+2=1+2+2=1+1+3$$ For a given number $n$,if we are trying to calculate ...
3
votes
1answer
282 views

For integer partitions, why does $e(n)-o(n)=k(n)$?

I denote by $e(n)$ is the number of partitions of $n$ with an even number of even parts, $o(n)$ the number of those with an odd number of even parts, and $k(n)$ the number of those that are ...
1
vote
1answer
396 views

Restricted Integer Partitions

Two Integer Partition Problems Let $P(n,k,m)$ be the number of partitions of $n$ into $k$ parts with all parts $\leq m$. So $P(10,3,4) = 2$, i.e., (4,4,2); (4,3,3). I need help proving the ...
4
votes
2answers
387 views

The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
2
votes
1answer
187 views

dual representations and partitions

Suppose we have a partition $\mu$ of $n$. There is an associated polynomial irreducible representation $\phi_{\mu}$ of $GL_n(\mathbb C)$. How do I obtain a new representation of $GL_n(\mathbb C)$ ...
31
votes
3answers
737 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
4
votes
3answers
462 views

Number of partitions of $2n$ with no element greater than $n$

The number of partitions of $2n$ into partitions with no element greater than $n$ (copied and slightly adapted from http://mathworld.wolfram.com/PartitionFunctionq.html), so I'm looking for a nice ...
7
votes
0answers
281 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
5
votes
2answers
212 views

Partitions in which no part is a square?

I asked a similar question earlier about partitions, and have a suspicion about another way to count partitions. Is it true that the number of partitions of $n$ in which each part $d$ is repeated ...
7
votes
2answers
875 views

Identity involving partitions of even and odd parts.

First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions ...
4
votes
2answers
889 views

Same number of partitions of a certain type?

Is there a quick explanation of why the number of partitions of $n$ such that no parts are divisible by $d$ is the same as the number of partitions of $n$ where no part is repeated $d$ or more times, ...
4
votes
5answers
2k views

Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$

How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$ And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$ In both cases, ...
1
vote
2answers
127 views

Every number $n^k$ can be written as a sum of $n$ distinct odd numbers

I wish to prove that for $n,k\in\mathbb{N} > 1$, we can always write $n^k$ as a sum of $n$ odd positive integers. I have an idea of how to approach this, but my method seems to cumbersome. I am ...
12
votes
4answers
2k views

Prime Partition

A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
2
votes
2answers
246 views

Number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s

How do I find the number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s? I think the title is self-explanatory. E.g., if I have to represent $13$ as a sum of only ...
1
vote
1answer
180 views

Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. For example, if a Jordan form consists of a single cell $2 \times ...
1
vote
1answer
409 views

The coin change problem in the quantitative way

Today,I came across this problem: Suppose you have a currency, named miso, in three denominations, $1, 10$ and $50$. In how many ways can $107$ miso be given in this currency if you have ...
2
votes
2answers
180 views

On the path diagram of a partition

A path diagram is a Ferrers diagram for a partition $\lambda$ in which the dots have been replaced with numbers computed as follows. The number in position $(i,j)$ the number of paths (going either ...
4
votes
3answers
189 views

A formula on partitions

Suppose that $\lambda,\mu$ are integer partitions, with conjugates $\lambda^*,\mu^*$. Could you help me to prove the following formula, please? ...
1
vote
2answers
458 views

Proving that exactly half the partitions of $n$ into powers of 2 have an even number of parts

Could you help me how to prove that exactly half the partitions of $n$ into powers of 2 have an even number of parts, please?
6
votes
2answers
263 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...
4
votes
1answer
409 views

Exponential generating function for restricted compositions

I wanted to know if it is possible to use exponential generating functions to evaluate composition of N using K distinct numbers (where the supply of numbers is infinite)? For e.g if N=10 and ...
2
votes
4answers
267 views

Summing up Problem (Combinations)

This is a part of a bigger problem I was solving. Problem: $N$ is a positive integer. There are $k$ number of other positive integers ($\le N$) In how many ways can you make $N$ by summing up any ...
3
votes
2answers
1k views

Number of ways to represent a number from a given set of numbers

I want to know in how many ways can we represent a number $x$ as a sum of numbers from a given set of numbers $\{a_1.a_2,a_3,...\}$. Each number can be taken more than once. For example, if $x=4$ and ...
2
votes
1answer
122 views

Partition function with restrictions

What is the number of ways of partitioning a positive number $k\leq mn$ using non-increasing parts such that the number of parts can be at most $n$ and the value of each part can be at most $m$?
3
votes
2answers
138 views

Infinitely many $n$ such that $p(n)$ is odd/even?

We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd. It might be proved ...
4
votes
2answers
558 views

Some other ways to prove a partitions problem

Show that the number of partitions of the integer $n$ into three parts equals the number of partitions of $2n$ into three parts of size $< n$. I can only prove it by building a bijection ...
5
votes
3answers
229 views

Two series relations, each one implies the other - from Andrews' partition book

That's my first question here, and i was encouraged to post because my question in MathOverflow (HERE) was beautifully and fast answered. But my questions in not at research level... As i said there, ...
4
votes
1answer
142 views

Partition and power of 2s

I'm trying to find a partition of natural number N whereby we can generate a partition for all natural numbers less than N from the partition elements. The number of elements in the partition should ...
4
votes
1answer
108 views

HAKMEM 18(B): Cubic Partitions

Taken from HAKMEM 18. Quoting... A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of ...
15
votes
2answers
285 views

A question on partitions of n

Let $P$ be the set of partitions of $n$. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
18
votes
3answers
556 views

An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$ B_n^2 \leq B_{n-1}B_{n+1} $$ for $n \geq ...
5
votes
2answers
374 views

How can I reduce a number?

I'm trying to work on a program and I think I've hit a math problem (if it's not, please let me know, sorry). Basically what I'm doing is taking a number and using a universe of numbers and I'm ...
1
vote
1answer
433 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
9
votes
1answer
574 views

Recurrence for the partition numbers

I'm reading Analytic Combinatorics [PDF] book by Flajolet and Sedgewick, and I can't figure out one of the steps in the derivation of the $P_n$ — number of partitions of size $n$ (or coefficients in ...
2
votes
2answers
223 views

Two combinatorics problems. I'm not 100% confident in my answers

These are two problems from my combinatorics assignment that I'm not quite confident in my answer. Am I thinking of these the right way? Problem 1: On rolling 16 dice. How many of the $6^{16}$ ...
4
votes
2answers
451 views

Learning about partitions and modular forms

I'm interested in learning about partitions and modular forms. I already know algebra and analysis (complex and real). Can any one suggest me books or other materials from where I can learn these ...
1
vote
0answers
161 views

Jordan Measure, Semi-closed sets and Partitions

From earlier question here. Consider $$\mu \left( (0,1) \cap \mathbb Q \right) = 0$$ where $$(0,1) = \left(0,\frac{1}{3}\right)\cup \left[\frac{1}{3}, \frac{2}{3}\right] \cup \left(\frac{2}{3}, ...
2
votes
2answers
405 views

Jordan Measures, Open sets, Closed sets and Semi-closed sets

I cannot understand: $$\bar{\mu} ( \{Q \cap (0,1) \} ) = 1$$ and (cannot understand this one particularly) $$\underline{\mu} ( \{Q \cap (0,1) \} ) = 0$$ where $Q$ is rational numbers, why? I ...
26
votes
1answer
868 views

Ellipse 3-partition: same area and perimeter

Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same ...
6
votes
1answer
4k views

The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts

This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give ...
3
votes
1answer
453 views

Partition number problem

Denote by $I_m=\{0,1,2,…m\}$, by $N_s=\{1,2,…,s\}$ , by $\overline s$ least common multiple of elements of set $N_s$ and by $p(k,N_s)$ the number of partitions of natural number $k$ in parts used ...
5
votes
1answer
296 views

Number of ways to sum square numbers to yield a given number

I would like to know how many choices of $x_i$ there are such that $$\sum_{i=1}^{n}x_i^2=m$$ where $n$, $m$ are given. The $x_i$ can be any nonnegative integer and need not be unique and the order is ...
0
votes
2answers
988 views

Partition Problem, verifying solution in polynomial time

I add a look at the partition problem, this problem is know as the Easiest hard problem since it is NP-complete and seems pretty easy. From wikipedia on NP-complete: In computational complexity ...
4
votes
1answer
2k views

Hardy Ramanujan Asymptotic Formula for the Partition Number

I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about partitions). The asymptotic formula always seems to be written as, $ p(n) \sim ...
1
vote
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
votes
2answers
552 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 $p(n)$ ...