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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73 views

If $P(n, k)$ is the number of partitions of $n$ elements into $k$ sets, then $P(n, k) = kP(n-1, k) + P(n-1, k-1)$ [on hold]

A partition of the set $\{1, 2, \dots , n\}$ into $k$ parts is a way of writing the set as a disjoint union of k subsets. For example $\{1, 2, 3, 4, 5\} = \{1, 4\} \cup \{2, 3\} \cup \{5\}$ is a ...
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63 views

Explain this generating function

I have a task: Explain equation: $$\prod_{n=1}^{\infty}(1+x^nz) = 1 + \sum_{n=m=1}^{\infty}\lambda(n,m)x^nz^m $$ $\lambda(n,m)$ - is number of breakdown $n$ to $m$ different numbers (>0) It's ...
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0answers
59 views

Integer partitions without rotated solutions?

I'm searching for an algorithm to determine a list of all integer partitions of a number $n$ into a fixed number $m$ of summands (say $n=6$ and $m=4$), for instance to be stored into a list of ...
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1answer
25 views

Growth rate of ordered bounded partitions

Let $P_n(k,i)=|\{(d_1,\cdots,d_i): \ \sum_{j=1}^id_j=n, \ \forall j\ 1\leq d_j\leq k\}|$, the number of ordered partitions of $n$ into $i$ parts with individual parts bounded by $k$, with no piece ...
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39 views

Partitions of $\mathbb{R}^+$ into subset closed by sum and product

Suppose we can partition $\mathbb{R}$ into two subset $A,B$, both non empty and closed by sum and product. Let $0\in A$, and suppose that exists $b\in B$. Then $b^2\in B$. Now, $-b\in B$, cause if ...
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37 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
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1answer
62 views

Count the number of ways n different-sided dice can add up to a given number

I am trying to find a way to count the number of ways n different-sided dice can add up to a given number. For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ...
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24 views

Partitioning a totally ordered set into three subsets according to the order

Consider a set $S$, and a total order $R$ over that set. Part (a) Given some element $e \in S$, explain why it is possible to partition $S$ into the following three sets: $$S_1 = \{ x \in S ...
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2answers
46 views

Geometric meaning of reflexive and symmetric relations

A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of ...
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1answer
61 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
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32 views

Arithmetic properties of the partition function

Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three" $p(5n+4)\equiv0 \mod 5$ $p(7n+5)\equiv0 ...
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1answer
41 views

partition with infinite entropy?

Let $P$ be an infinite partition of the interval $[0,1]$. Let $P$ have elements $I_i$ which has Lebesgue measure $m(I_i)$. Then the entropy of $P$ is defined by $\sum_i -m(I_i)\log m(I_i)$. Can this ...
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1answer
86 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 ...
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0answers
33 views

How many partitions of 12 that fit the requirements?

How many partitions of $12$ are there that have at least four parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to $4,3,2,1$? ...
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1answer
49 views

Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, ...
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135 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 ...
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72 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
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16 views

asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
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1answer
59 views

Partitions applications in physics

Is there any direct application of all developments related to partitions? I am specially interested in physics but cryptography or other mostly theoretical areas would also be a good answer. By ...
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1answer
52 views

Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
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1answer
35 views

Partitioning into groups with maximal mixing

Suppose I have a class of 30 students and I want to give them 8 assignments to do in groups of 3. As far as possible I'd like the students to work with as many different students as possible. Ideally ...
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1answer
36 views

constant length of blocks in partitions

Let's assume we have partitions $P_k$ of the set $\{1,...,n\}$. If we choose two partitions it can happen, that each of them has a constant length of its blocks, but that the intersection of these two ...
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44 views

Difficulty parsing combinatorics exercise

I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head. (2.2.8) ~ Let ...
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25 views

Bound and parity integer-partition into fixed number of parts

Let $E(n,k)$ denote how many set of $k$ distinct non-negative integer are there such that their sum is an even number $\leq n$. Let $O(n,k)$ denote how many set of $k$ distinct non-negative integer ...
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47 views

Number of set partitions of n elements into k sets with subsets of size r not allowed

This is a generalization of the question Number of ways to partition a set with $n$ elements to $k$ subsets where at least one subset has $r$ elements . At the end of answer for this question, there ...
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1answer
55 views

Compactness of a set of partitions

The interval $[0,1]$ is partitioned to $n$ disjoint parts. Is the set of all possible partitions compact? There are several cases: A. All $n$ parts are connected intervals (possibly empty). In this ...
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2answers
57 views

What does George Andrews mean by i in “Theory of Partitions”?

From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck): What do these last two sentences mean? I don't get "where exactly $f_l$ of the ...
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26 views

An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
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35 views

How to calculate least moves of fruit.

I have a question, I'll try to abstract from the real problem to not lose people. What I'm really looking for is the name of the algorithm or class of problem to find my solution. I feel that this is ...
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39 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
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52 views

How to find Partition of unity in $ \mathbb{S}^n$ with only $2$ functions

How to find Partition of unity in $\mathbb{S}^n$ with only $2$ functions?
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83 views

Ways of Distributing $n$ balls among $k$ boxes, each box containing $L \leq x_i \leq M$ or $0$ Balls

I need to calculate the number of ways of distributing $n$ balls among $k$ boxes, each box may contain no ball, but if it contains any, then it must contain $\geq L$ & $\leq M$ balls. This ...
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42 views

intersection of stabilizers of block systems

Lets assume the following situation: $G$ acts regularly on a set $M$. Then there is a bijection between the set of subgroups and the set of blocks containing a fixed element $m \in M$. The blocks ...
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1answer
26 views

Example where a cover is fewer than a partition?

Given a Cartesian product of sets $X \times Y$, A (combinatorial) rectangle is a set $A \times B$ where $A \subseteq X$ and $B \subseteq Y$. Given a function $f : X \times Y \rightarrow \{ 0, 1\}$ one ...
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23 views

generalization of bijection proof

$p(n|\text{odd parts}) = p(n|\text{distinct parts})$ I need bijection proof and generalization of this proof. for $k = 7$ odd parts are : ...
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61 views

OEIS sequence A086449

OEIS sequence A086449 http://oeis.org/A086449 is defined by: $a(0)=1$, $a(2n+1)=a(n)$, $a(2n) = a(n)+a(n-1)+\ldots+a(n-2^m)+\ldots$ $= a(n)+\sum_{i=0}^{\lfloor\lg n\rfloor}a(n-2^i)$ One can show ...
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1answer
48 views

Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
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1answer
69 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
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1answer
33 views

A bound on the balanced equipartition of a multi-set of integers

A balanced equipartition of a multi-set of $2n$ integers is a partition into two multi-sets $S_1$ and $S_2$ of size $n$ such that the sum of the integers in $S_1$ is as close as possible as the sum of ...
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55 views

Generation of n terms that sums to a specific value?

I'm about to conduct an integer program where I am to transport some products from different cities to docks and refinement stations. I am however required to work with simulated data where I only ...
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1answer
25 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
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What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?

Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with ...
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192 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
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1answer
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Ramsey Theory: Showing the existence of a special set of natural numbers.

Show that there is an infinite set of natural numbers such that the sum of any two elements has an even number of prime factors. My attempt: Define a coloring on the doubletons of $\mathbb{N}$, that ...
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67 views

How to prove this identity?(perhaps related to partition)

How to prove this identity? $$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)} = \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$$ Maybe the method using generating functions is good.
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24 views

Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
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1answer
98 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
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31 views

Cauchy's Coefficient Formula

How can I recover the coefficients of a polynomial power expansion of the type (number of partitions): $ \prod_i (1+x^{\alpha_i}z) = A_1 z + A_2 z^2 + A_3 z^3 + A_4 z^4 \ldots $ using Cauchy's ...
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77 views

Finding the amount partitions of a multiset

A multiset $A$ contains $n$ positive integers. The multiplicity of every integer is less or equal to $m$. $A$ is partitioned into $m$ subsequences in such a way that the multiplicity of all elements ...
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1answer
51 views

Number of ways to add up to a number without repetition (order does not matter)?

I have a number x and want to find how many ways there are to add up to that number using the y numbers from numbers 1-z. for example, for x=15 y=3, z=9, there are 8 ways to add up to 15 using 3 ...