# Tagged Questions

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 number of balanced two-way partition of the set

Given a set with $2n$ elements, show that the number of balanced two-way partition of the set $$P(2n)=\frac{2n!}{2\times n!\times n!}$$ I'm getting is as P(2n)=${2n}\choose{n}$. But I'm getting ...
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### Algorithm for generating integer partitions

I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate all of them and then discard the ones that ...
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### Partition of $S = \{1,2,\dots, 3n\}$ in to three subsets $A, B, C$ such that $|A| = |B| = |C| = n$

Let $n$ be a positive integer and consider the set $S = \{1,2,\dots ,3n\}$. Show that, for every partition of $S$ into 3 subsets $A, B, C$ such that: i) $|A| = |B| = |C| = n$ (here $|X|$ denotes the ...
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### On partitions of integers

In an example in my textbook, I came across a question where it was asked to find the generating function for the number of partitions of ${n \in N}$ into summands that (a) cannot occur more than 5 ...
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### Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product

Problem: Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product My attempt: This problem would be solved if it could be proven that 9 consecutive ...
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### On partition of integers

I came across an example in my textbook where it was asked to find the generating function for the number of integer solutions of: ${2w+3x+5y+7z=n}$ where ${0\le w, 4\le x,y, 5\le z}$ The ...
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### Is there a formula for the number of equipartitions of $[n]$ into $k$ parts of size $s = n/k$?

Let $k$ divide $n$ and $Q(n,k)$ be the number of partitions of $n$ into $k$ parts, each of which has size $s = n/k$. Is there a formula for $Q(n,k)$? What is the asymptotic behavior of $Q(n,k(n))$ ...
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### Counting integer partitions of n into exactly k distinct parts size at most 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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I've been given an integer partition A = (A1, A2, ... ,An) and its conjugate B = (B1, B2, ... ,Bm). Using that information, I'm tasked with proving that $$\sum_{i=0}^n (i-1)A_i = \sum_{j=1}^m B_k(B_k ... 1answer 95 views ### How to prove that we can increase the precision of Riemann sum if we refine a partition When learning Riemann Integral, I was introduced to the concept of partitions and refining them. It stated that refining a partition (e.g. controlling its norm) increases its efficiency. What I mean ... 1answer 17 views ### Show that the following Set \Lambda is a Partition Consider the two sets$$A_r = \{x \in \mathbb{Z} \enspace | \enspace x \enspace = \enspace 5q + r, \enspace 0 \enspace \leq \enspace r \enspace < \enspace 5\}\Lambda = \{A_r\}$$We must ... 0answers 31 views ### Lower bounds/upper bounds for Qbinomials Is there any lower bound or upper bound known for Q-binomials? I know that number of partitions function p(n)>2^(\sqrt n). But, I don't know any lower bounds for Q-binomials which are the generating ... 1answer 25 views ### Partitioning a Set: Need help with Notation. If I have a relation:$$R=\{(x,y)\in\mathbb R^2 : \cos(x)=\cos(y)\},$$it is clear to me that [x]= \{x,-x : x \in \mathbb R\} What I'm trying to say is that the equivalence class x is partitioned ... 1answer 69 views ### Proof of an identity for n! involving integer partitions of n Let B(n) be the set of the integer partitions of the integer n\gt0, with the notation:$$B(n)=\left\{(b_1,\ldots,b_n)\in\mathbb{N}^n \ \ ; \sum_{i=1}^{n} i\cdot b_i=n \right\}$$... 0answers 32 views ### Partition Problem with discontiguous sets I'm trying to solve a variant of the partition problem. I have two important twists. I need to solve for k partitions, not just 2, as in the classic partition problem. The following code does that: ... 1answer 46 views ### Can you partition a rectangle into exactly 3 congruent non-rectangular parts? Recently I came upon the following result: Theorem (*): Let n be a positive integer not equal to 1,3,5,7,9. Then it is possible to partition a rectangle into exactly n congruent non-rectangular ... 1answer 26 views ### Determining the number of different ways a partition can occur? Say for instance a food stall keeps a record of how many complaints it receives during its 4 day work week. Complaints are classified as Mon, Tues, Wed or Thurs, depending on the day during the week ... 0answers 16 views ### How to construct a partition of X given a sigma-algebra on X (when X countably infinite) [duplicate] I am attempting to construct a partition of a countable set X, given a sigma-algebra on that set. Eventually I will want partitions of X to be in 1-1 correspondence with sigma-algebras on X. See my ... 0answers 89 views ### Closed form for \sum_{n=1}^\infty \frac{1}{P(n)}, where P(n) is the partition function. Is there a closed form for the following infinite series?$$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$where P(n) is the partition function. 1answer 42 views ### Is this integer partition studied? First, this is my firt post here; I'm not a rigorously trained mathematician so apologies for the abuse of language, or if the problem is too trivial :) In a finite-system problem in statistical ... 1answer 67 views ### Maximizing the sum of exponentials whose exponents sum to N Let N \geq 1 be a sufficiently large integer, let a > 1 be a real number, and let n_1, \dots, n_t be integers between 0 and K, where K divides N. I want to determine the following:$$...
Fix a positive integer $k$. For positive integer $n$ let $p(n;\le k)$ denote the number of partitions of $n$ into at most $k$ parts, and let $p(0;\le k)=1$. (1) Show that there is a polynomial $P(x)$ ...