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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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 ...
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partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
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Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
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explicit random cake cutting

I like to split a given interval, let's say $[0,1]$, randomly to a given number $n$ parts. A random input may be provided, like for example a sequence of random numbers $\omega=(r1, r2, ...
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36 views

Partitions that have at most k parts and all parts <= j

Let p¯k(n) be the number of partitions of n with largest part at most k which is equivalent to partition into at most k parts. I do know an expression for that function. ( product of 1/1(1-n) through ...
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Teminology: Partitioning a Set Including Empty Partitions

Mostly I see a partition of a set A defined as a collection of non-empty disjoint sets whose union is A. I see one reference that allows empty sets to be included in the partition: ("Potter, M. "Set ...
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Discriminating integer partitions

Given a fixed positive integer, say $n$, letn $P_n$ be the set all partitions of $n$, where each partition itself is a set i.e. order is discarded and each part is less than 5. Can we establish a ...
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63 views

Running the Greene-Nijenhuis algorithm backwards

Let $Y$ be a Young tableau of shape $\lambda:=(\lambda_1,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n$ and $N:=|\lambda|:=\sum_i\lambda_i$. I'll be using the English ...
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56 views

Non-standard partition of integers question

The question is as follows. Partition an integer $n$ into $r$ distintc parts with each part ranges from $[1,m]$ and the parts order is irrelevant. How many ways of different partitions are there? ...
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37 views

What is the big-$\mathcal{O}$ bound for the sum of function applied to the partitions of a set?

Consider a set $A$ that is partitioned into $n$ subsets $A_1 | A_2 | ... | A_n$ and a function $f \in \mathcal{O}(g)$. Question: what is the tightest bound I can establish for $\sum_{i=1}^n ...
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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 ...
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Number of solutions (excluding permutations of variables' values) and solving in distinct positive integers the following system of equations

Questions and important info in italics, very important ones in bold. Here we have the system; $V_{1}+V_{2}\cdots+V_{k}=A$ and $V_{1}^{2}+V_{2}^{2}+\cdots +V_{k}^{2}=B$ where $V_{1}$, $V_{2}$, etc. ...
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Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda]) $, where ...
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Given $x = \sum_{i \in k} a_i$ from $\{ a_1, \ldots, a_n \}$, partition $x$ as $\sum_{j \in k} a_j$

Let me put this into an example. We have an array of the first $r$ (example $r=7$) natural numbers; $1, 2, 3, 4, 5, 6, 7$ and given $k=3$, so you get to select any three integers from the pool ...
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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}, ...
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49 views

Number of Sets of Partions

I looked at the partitions of numbers, like let's say $n=5$. You get $$ \begin{eqnarray} 5&=&5\\ \hline &=&4+1\\ &=&3+2\\ \hline &=&3+1+1\\ &=&1+2+2\\ \hline ...
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33 views

Minimize sum of squared error

I have an array of real numbers, I want to partition them into k sets. In each set, I calculate the sum of squared error. Then, I add up all the sum of squared error for all the set. I want to ...
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30 views

Partition problem of equal size

I have an array S of size 2n, each element in the array is an integer I want to split it into two arrays of size n, and under this condition, minimize the difference of the sum of integers in the two ...
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29 views

Partition partition with constraint of equal size

I see the problem here Polynomial complexity algorithm of partition problem with sets of equal size This is the well know partition problem but with constraint that the size of both sets must be ...
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34 views

number of pairs that are member of a class of one of the given partitions

Suppose I am given a set of S partition of universal set A, define pair as (a,b) iff elements a,b are members of one of set in at least one partition. I want to know how many pairs in S. Example: $$A ...
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What are the number of “minimal 2-complete” partitions for the first 100 natural numbers?

Let $λ = (λ_0, λ_1, . . , λ_n)$ be a partition of the natural number $m$ into $n+1$ parts $λ_i$ such that: 1) $m = λ_0+ λ_1+ . . . . + λ_n$ 2) $λ_0 ≤ λ_1 ≤ . . . . ≤ λ_n$ The partition $λ = ...
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33 views

Partitions of numbers

It needs to prove that the partitions of n, in which all the terms do not exceed k, so same number of partitions of n into at most k non-zero terms. I tried to prove it by using Stirling numbers of ...
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17 views

K-way Undirected Weighted Graph Partition with K Vertices Pre-Assigned

I have an undirected weighted graph to be partitioned into k subgraphs with minimal edge weight between the partitions and k of the vertices are constrained to lie in separate partitions. I am ...
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81 views

Real Analysis: Show that g is integrable on [a,b] and that $\int_a^b$ $g(x)dx=$ $\int_a^b$ $f(x)dx$

Suppose f is integrable and g is bounded on [a,b], and g differs from f only at points in a set H with the following property: For each $\epsilon>0$, H can be covered by a finite number of closed ...
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35 views

Special partition of a number $n$

Given any integer $n$, how many ways can it be partitioned in which the number $1$ is not allowed? For instance, if $n=6$, then the partitions obeying the aforementioned rule are $6+0$, $4+2$, $3+3$, ...
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27 views

Ferrers Graph question

Use Ferrers diagram to show that the number of partitions of an integer into parts of even size is equal to the number of partitions into parts such that each part occurs an even number of times ...
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21 views

Order on the set of partitions (terminology)

Let $S$ and $T$ be partitions of some set $U$. What is the name for the partition $\{ X\cap Y \mid X\in S, Y\in T, X\cap Y\ne\emptyset \}$? Should it be called the infimum of $S$ and $T$? meet of ...
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37 views

A lattice generated by two particular sublattices of the lattice of binary relations

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y_X\in\mathscr{P}U$ for every $X\in S$ (that is $Y$ is a ...
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79 views

Another conjecture about filters and cartesian products

Now I present a similar but different (weaker) question (the difference is a different definition of the set $\Gamma$): Let $U$ be some set. Let $\Gamma$ be the set of all finite unions ...
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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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115 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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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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38 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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45 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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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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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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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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How many partitions of $N$ are there into $n$ non-negative parts $c_k$ such that $\sum_{k=1}^n c_k = N$ and $\sum_{k=1}^n kc_k = M$??

So when coming up with a recursive solution to a counting problem of placing 1's into an $N \times N$ matrix ($N$ even) so that every row and every column has exactly $N/2$ 1's, my recursive ...
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29 views

Mapping a set of sets to a partitioning.

I've been experimenting with the following idea, and I wondered if there's a name for it: Suppose $S_0, S_1, ... S_{n-1}$ is an array of $n$ sets of elements in $U$. Now for any element $e \in U$ we ...
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26 views

Partition of the node set of a graph into connected subsets

What word is most commonly used in graph theory for a partition of the node set of an undirected graph into connected subsets? More rigorously: Given an undirected graph $(V,E)$, a partition $S ...
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37 views

Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
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How to prove this identity about Sylvestered partitions of n into m parts such that …

Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ ...
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Let $W_1(r,m,n)$ denote the number of partitions of n into m parts ,each larger than 1, with exactly r odd parts,each distinct.

Let $W_1(r,m,n)$ denote the number of partitions of n into m parts ,each larger than 1, with exactly r odd parts,each distinct. Let $W_2(r,m,n)$ denote the number of partitions of n with $2m$ as ...
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40 views

Prove the set of m residue classes cover the integers

How would I show that the set $\{k+qm:q\in \mathbb{Z}\}$, where $k=0,1,2,...,m-1$ covers the integers? I'm thinking to assume there exists such an integer and then derive a contradiction but I'm not ...
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Partition of $[a,b]\subset\mathbb{R}$

Is it possible to create a numerable partition of $[a,b]$? Because I think that it isn't possible, because the last point of the partition must be $b$. But I use the function defined in $[0,1]$ that: ...
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The Generating Functions for $p(S_d,n)$

Prove: $$\sum_{n=0}^{\infty}p(S_2,n)q^n=\prod_{j=1}^{\infty}\frac{1}{(1-q^{5j-4})(1-q^{5j-1})}$$ I have been trying to figure out where to even start this problem and I have no idea what to do. Can ...
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Partitions of a closed interval on the reals

I'm currently trying to go through my textbook in real analysis where the integral is defined. And I'm really confused by something that seems very counter intuitive, and the proof isn't given, and so ...
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67 views

A Partition Generated from a Family of Sets

This assertion is so basic that I’d expect it to have been put forward with someone’s name attached 100 years ago. But, I can’t find any reference to it searching the web. Of course, the other ...
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Need help with a Partitions question

For $m \in \mathbb{N}$, let $C_m = \{x \in R \mid m-1 \leq x^2 < m\}$. Is $\ell=\{C_m \mid m \in \mathbb{N}\}$ a partition of $\mathbb{R}$? If I understand it correctly, $C_m$ will always be a ...
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325 views

Number of ways to divide a stick of integer length $N$, take 2

This is a follow up and motivated by this question, Number of ways to divide a stick of integer length $N$, Suppose we have a stick of integer length $N$. I'm looking for (preferably closed-form) ...