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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6
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1answer
165 views

Representations of an integer as the sum of other integers

Given a finite set $S$ of (distinct) integers $s_1, \dots, s_n$ and an integer $x$, I'm looking for all representations (where order is important) $$ x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\}) $$ ...
4
votes
1answer
109 views

How to extract coefficient of $x^n$ in an infinite product generating function?

Are there methods for obtaining the coefficient of $x^n$ in a generating function like $$\prod_{i=1}^\infty Q(x),$$ where $Q(x)$ is a rational function? This arises when we want to count partitions of ...
4
votes
1answer
74 views

Partitions of a prime power into powers of the same prime

Fix a prime $p$, and $k$ a natural number. The question is then: How many partitions of $p^k$ are there into powers of $p$? So, for instance, if $p = 2$ and $k = 2$, there are 4, namely (4), (2, 2), ...
1
vote
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: $a(0)=1$, ...
1
vote
1answer
25 views

List 4 partitions of the alphabet containing 3 sets. How many different partitions can be made?

The first part of the question is easy; there are many right answers but I put down: $$(a), (b), (c..z)$$ $$(a,b), (c), (d..z)$$ $$(a,b,c), (d), (e..z)$$ $$(a,b,c,d), (e), (f..z)$$ The part I am not ...
1
vote
1answer
34 views

Are values of multinomials distinct for distinct sets of integer partitions in the denominator?

Let a multinomial be denoted by $$M(n, K) = {n! \over {\prod k_j!}}$$ where $K= (k_1, k_2, ..., k_n)$ and $k_1 \ge k_2 \ge ... \ge k_n$. It is obvious that K is an integer partition of n. Then, my ...
1
vote
1answer
88 views

Question on lower & upper Riemann sums for piecewise.

Partition{$\frac{-\pi}{6},3,2\pi$} $f(x)= 1/2$ if x is rational $sin(x)$ if x is irrational. Im not sure if im doing it correctly: ...
0
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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, ...
0
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1answer
57 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 ...
0
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1answer
25 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 ...
0
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1answer
16 views

Regular expression for a particular language

Several years ago I came across a paper that defined a regular expression (or collection of regular expressions?) for a specific language. The language is the language of set partitions enumerated by ...
0
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1answer
32 views

Finding Partition, Riemanns Integral

Define $f:[0,2]\rightarrow\mathbb{R}$ by setting $f(x)=1$ if $x\not=1$ and $f(1)=3$. Find a partition $D$ of $[0,2]$ for which $S_D-s_D<2^{-1000}$.
0
votes
1answer
24 views

Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r)

Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r) or odd and congruent to 2r-1 or 4r-1(mod 4r). Let $P_2(r;n)$ denote the number of ...
0
votes
1answer
26 views

Do cosets of this semigroup partition this ring?

Let $R = \Bbb{Z}^3$ be the usual direct product ring, and let $S = \{ (k^a, k^b, k^c) : k \in \Bbb{Z} - \{0\}\}$ be a subsemigroup of $R$. Then do the cosets $aS$ partition $R$? Let $R^* = S^{-1}R$ ...
0
votes
1answer
29 views

Partition set to contain the same number of elements distributing the remainder

Given $|B| = 23$ and number of partitions $P=4$. We want to partition the given set $B = B_1\cup\dots\cup B_P$ so that every partition $B_i$ contains the same amount of elements where the remaining ...
0
votes
1answer
75 views

How to find random numbers that can sum up to n?

I have a random integer $n$ and another integer called the summary. I want to know how many ways I can sum a subset of numbers from $1$ to $n$ to produce the value of summary. For example, I have ...
-1
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1answer
113 views

I'm taking an advanced math paper and I have no idea how to start this question!

How would I go about working this out? I honestly don't know where to start! Any help is appreciated.
0
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0answers
43 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 ...
0
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0answers
21 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 ...
0
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0answers
38 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 ...
0
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0answers
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 ...
0
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0answers
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 ...
0
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0answers
40 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 ...
0
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0answers
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 : ...
0
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0answers
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 ...
0
votes
0answers
22 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 ...
0
votes
0answers
10 views

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 ...
0
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0answers
27 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 ...
0
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0answers
21 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 ...
0
votes
0answers
35 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 ...
0
votes
0answers
18 views

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 $S(T\,;m,\,n)$ ...
0
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0answers
20 views

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 ...
0
votes
0answers
37 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 ...
0
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0answers
216 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 ...
0
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0answers
40 views

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: ...
0
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0answers
38 views

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 ...
0
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0answers
28 views

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 ...
0
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0answers
51 views

Refinements and partitions

Suppose $P_1, ... P_n $ are finite partitions of the set $X$. Let $Q_n$ be the smallest common refinenement of the partitions $( P_i ) $. I want to show that $Q_{n+1}$ is a refinement of $Q_n$. My ...
0
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0answers
60 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 ...
0
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0answers
53 views

The generating function for partitions: an alternate (and false) representation

The combinatorics book I'm going through asked me to find the generating function whose coefficients $p_n$ give the number of integer partitions of $n$. I reasoned the following answer: the generating ...
0
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0answers
24 views

Tournament payouts, or more number partitioning/change-making

I was thinking of this when confronted with the problem of tournament payouts... Let's say I have \$100 to be parceled out to 10 different people, each of whom is due $n_1, n_2,..., n_{10}$ dollars. ...
0
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0answers
47 views

Number of Singleton Blocks in Set Partition

I'm interested in some general information on the following question: Consider the collection of partitions of an $n$-set into $m$ blocks as a uniform probability space. Let $X$ be the random ...
0
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0answers
32 views

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 ...
0
votes
0answers
36 views

The number of 2's in all partitions of n

Let $a_{n}$ be the number of "2"'s that appear in all partitions of $n\geq 0$. The sequence begins like $(0,0,1,1,3,4,\cdots)$. I am tasked to show that it's OGF is $$\dfrac{x^{2}P(x)}{(1-x)^{2}}$$ ...
0
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0answers
11 views

Approximations for the Coarse Graining of the one norm difference of two probability distributions

I want to coarse grain $D(P_{1},P_{2}) = \frac{1}{2} \sum_{r}^{D} |Pr(r|1) - Pr(r|2) |$ for two distinct distributions Pr(r|0) and Pr(r|1). Such that $\sum_{r} P(r|1) = 1$ and $\sum_{r} P(r|2) = 1$. ...
0
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0answers
279 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) ...
0
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0answers
92 views

Knapsack like problem for product of distinct primes

While looking into ways of generating certain kinds of pseudo-random number sequences I came up with the issue of finding the maximum of products of distict primes with a sum less than N. I'm ...
0
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0answers
163 views

Give an expression for lower and upper sum, concluding $f$ is integrable

$f : [a,b] \rightarrow \mathbb{R}$ is non-increasing, which means that $f(y) \le f(x)$ when $y>x$ The two questions are : Give an explicit expression (without infima and suprema) of ...
0
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0answers
121 views

Matrix & Partition & Natural Number & Pattern

I would like to know if someone know, how is called a matrix M*N, where m represents the row index in the matrix and the sum of the N columns at this row. Meaning that each row represents a possible ...
0
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0answers
55 views

Prove that single machine can reduce from 2-partition problem

I'm given the following problem: Prove that single machine \sum U_i (number of tardy job) with release date constraints problem can reduce from 2-partition problem ...