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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14
votes
7answers
6k 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 ...
6
votes
2answers
716 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
718 views

Keep getting generating function wrong (making change for a dollar) [duplicate]

Possible Duplicate: Making Change for a Dollar (and other number partitioning problems) I am working on the classic coin problem where I would like to calculate the number of ways to make ...
3
votes
1answer
205 views

Bell number. Combinatorial proof.

$B_m$ is the Bell's number (the # of all partitions of the set $m$) and $B_m^*$ is the number of partitions of set $m$ with no singleton block We need to prove that $B_m = B_m^* + B_{m+1}^*$. ...
3
votes
1answer
3k 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 ...
11
votes
3answers
776 views

Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its ...
5
votes
2answers
336 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 ...
3
votes
2answers
723 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, ...
12
votes
5answers
360 views

How many sets of distinct non-negative solutions are there to $k_1+\cdots+k_n=k$?

How many distinct $n$-tuples with distinct non-negative integer elements are there that add to $k$. For example there are $6$ triples that add to $4$. Namely $(0, 1, 3)$ and its $6$ permutations. ...
11
votes
4answers
1k 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 ...
4
votes
2answers
1k views

Partitioning a natural number $n$ in order to get the maximum product sequence of its addends

Sorry if i ask this question. probably it's already answered somewhere else but i didn't find it. Suppose to have a natural number $n \ge 0$. Given natural numbers $\alpha_1,\ldots,\alpha_k$ such ...
18
votes
3answers
510 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 ...
4
votes
2answers
307 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 ...
4
votes
3answers
441 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 ...
4
votes
2answers
536 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 ...
2
votes
3answers
2k views

The number of ways to write a positive integer as the sum of distinct parts with a fixed length

I am a topologist and not terribly familiar with the combo literature so please forgive me if this is standard. I'm hoping for some sort of reference for this. Given a positive integer $n$, I wish ...
2
votes
2answers
364 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 ...
1
vote
0answers
119 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
4
votes
3answers
1k views

Number of cycles of all even permutations of $[n]$ and number of cycles of all odd permutations differ by $(-1)^n (n-2)!$

I'm trying to solve task 44 of the first chapter of Stanleys Enumerative Combinatorics (found here). Show that the total number of cycles of all even permutations of $[n]$ and the total number ...
3
votes
1answer
150 views

Upper and lower integration inequality

I would like to learn how to prove that the following inequality holds. Let $F$ be a bounded function on an interval $[a,b]$, so that there exists $B\geq 0$ such that $|f(x)| \leq B$ for every $x\in ...
3
votes
1answer
745 views

Partition an integer $n$ into exactly $k$ distinct parts

I know how to find the number of partition into distinct parts, which is necessarily equal to the number of ways to divide a number into odd parts. I also know how to partition n into exactly k parts. ...
2
votes
1answer
784 views

Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
2
votes
2answers
390 views

Proof of a Proposition on Partitions and Equivalence Classes

I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like, Proposition: If $\{A_i|i\in I\}$ is a partition of ...
1
vote
1answer
97 views

Help understanding solution to growth of partition function

I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated. Question Let the number of ...
1
vote
3answers
383 views

Proof of the duality of the dominance order on partitions

Could anyone provide me with a nice proof that the dominance order $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau ...
0
votes
1answer
178 views

Total number of solutions of an equation

What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, ...
25
votes
1answer
832 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 ...
10
votes
1answer
218 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.) Question. Is ...
7
votes
0answers
179 views

Partitions of 13 and 14 into either four or five smaller integers

There are exactly 18 partitions of the integer 13 into 4 parts, as on the left of the table, and also 18 partitions into 5 parts, as on the right of the table: $$\begin{array}{c|c} 10+1+1+1 & ...
2
votes
1answer
185 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
11
votes
6answers
678 views

Can every infinite set be divided into pairwise disjoint subsets of size $n\in\mathbb{N}$?

Let $S$ be an infinite set and $n$ be a natural number. Does there exist partition of $S$ in which each subset has size $n$? This is pretty easy to do for countable sets. Is it true for ...
4
votes
1answer
1k 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
151 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
1
vote
3answers
356 views

Frobenius coin problem

Suppose that you only have coins worth, say 3 and 5 euros. According to Sylvester result we can find the Frobenius nr $g(3,5)=15-3-5=7$ so 7 is the largest integer that cannot be written as ...
9
votes
1answer
438 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 ...
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 ...
5
votes
3answers
127 views

Counting problem: generating function using partitions of odd numbers and permuting them

We have building blocks of the following lengths: $3, 5, 7, 9, 11$ and so on, including all other odd numbers other than $1$. Each length is available in two colors, red and blue. For a given number ...
5
votes
1answer
798 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 ...
4
votes
1answer
95 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
4
votes
3answers
490 views

Partitions of $n$ into distinct odd and even parts proof

Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts. How do I ...
4
votes
1answer
404 views

In how many ways can you split a string of length n such that every substring has length at least m?

Suppose you have a string of length 7 (abcdefg) and you want to split this string in substrings of length at least 2. The full enumation of the possibilities is the following: ...
4
votes
1answer
392 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 ...
3
votes
2answers
128 views

a formal name of this partition

Consider a set $A=\{1,2,3,4,5\}$, is there any terminology for the following partitions of $A$ ? (1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$ (2) $A=\{\{1,2,3,4,5\}\}$.
2
votes
1answer
60 views

(i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for $\pi_m(n)$.

Questions: $\pi_m(n)$ is defined as the number of partitions of n in which each part is no larger than m. (i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for ...
2
votes
1answer
37 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
1
vote
0answers
305 views

Counting distinct restricted integer partitions of $n$ into exactly $k$ distinct parts less or equal then $M$

How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$? The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
1
vote
4answers
1k views

How many solutions does the equation $x_1 + x_2 + x_3 = 11$ have, where $x_1, x_2, x_3$ are nonnegative integers?

Help me understand problems of this type a bit more intuitively. The solution $C(3+11−1,11)$ seems simple enough, but I got stuck thinking about how many integers you are choosing from within $x_1$, ...
1
vote
0answers
155 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}, ...
1
vote
3answers
214 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\\ ...
4
votes
2answers
155 views

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...