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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Partition of integer with size constraint

A rather straightforward combinatorial question: Given numbers $X, q, n$ such that $0 \leq X \leq n(q-1)$, what are the total number of ways to express $X$ as sum of $n$ numbers, where each summand ...
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3answers
113 views

Combination Question

I currently have an open question about counting the possible ways of summing numbers. I am still exploring all the ideas provided - those within my level of understanding. This is a question ...
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1answer
166 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) $$ x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\}) $$ ...
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2answers
131 views

Number of partitions of an $n$-element set into $k$ classes

A partition of a set $S$ is formed by disjoint, nonempty subsets of $S$ whose union is $S$. For example, $\{\{1,3,5\},\{2\},\{4,6\}\}$ is a partition of the set $T=\{1,2,3,4,5,6\}$ consisting of ...
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1answer
61 views

Partitions of n with certain conditions

Let $p$ be prime and $n$ be any integer. Suppose $t=(n^{a_n}, \dots, 2^{a_2}, 1^{a_1}) \vdash n$, (i.e. $t$ is a partition of $n$, where we group repeated integers, so, for example, $2^{a_2}$ means ...
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0answers
235 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 ...
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2answers
324 views

Proving a relation between 2 sets as antisymmetric

Let $U = \{1,...,n\}$ And let $A$ and $B$ be partitions of the set $U$ such that: $\bigcup A = \bigcup B = U$ and $|A|=s, |B|=t$ Let's define a relation between the sets $A$ and $B$ as follows: $B ...
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2answers
73 views

Prove $\int \limits_0^b x^3$ = $\frac{b^4}{4} $ by considering partitions [0, b] in $n$ equal subinvtervals.

Hi Guys I was given this question as an exercise in real analysis class. Here is what I came up with. Any help is appreciated! Prove $\int \limits_0^b x^3$ = $\frac{b^4}{4} $ by considering ...
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0answers
20 views

Canonical partition function identity

The grand canonical partition function, Z, is given by $$ Z = \sum_{M=0}^\infty G(M)z^M =\frac{V_0(z)U_L(z)}{1−U(z)V (z)} \tag{1} $$ where G(M) is the canonical partition function of a chain of ...
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1answer
46 views

proving antisymmetry of partition refinement

Suppose $P$ is the set of all partitions of some set $S$. $R$ is a binary relation on $P$, the refinement relation, defined as $(\Pi_1,\Pi_2) \in R $ if and only if for every $S_1 \in \Pi_1$, there ...
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2answers
57 views

mesh of the partition $P = \{1,3,4,5\}$

Consider the interval $[1, 5]$ and the partition $P = \{1, 3, 4, 5\}$, what is the mesh of this partition? So i'd say the mesh of this is $\text{mesh}(P) = \max{(|P_{i} - P_{i - 1}|)}$ for $i = 1, ...
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0answers
57 views

Asymptotics of partitions in at most n parts, bounded by r

For every positive integers $n,r,w$ define $$ p_w(n,r)=\#\{ (i_1,...,i_r) | \, 0\leq i_1 \leq \dots \leq i_r\leq n, \, i_1+\dots+i_r=w\} $$ as the number of partitions of $w$ in at most $r$ piece ...
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1answer
97 views

Number of 2n-1 equal size partitions up to symmetry

Consider the $K_{2n}$ (or just the set $\{1,\dots,2n\}$) with $S_{2n}$ acting on the vertices. Moreover consider a collection of 2n-1 partitions of the vertices into two equal sized sets (repeated ...
4
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1answer
80 views

Which positive integers can be written in the following form?

I was investigating a generalisation of this problem and found that it reduced to finding where the expression $$\frac{p(p+2m+1)}{2}$$ is an integer, where $p\ge 2$ and $m \ge 0$. Since exactly one of ...
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0answers
46 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
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1answer
45 views

Unordered multinomial coefficients

Let $\{n_1,\ldots,n_k\}$ be a partition of the integer $m$, that is $m=n_1+\ldots+n_k$, and denote by $\mathcal{P}_m$ the set of all such partitions. For a partition $\pi\in\mathcal{P}_m$, the ...
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1answer
36 views

Easy set partition problem

So I have this problem : Let A be a non empty set an P1 and P2 be two random partitions of the set A. Prove that the set is also a partition of A. I know that this is probably very easy to most of ...
2
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1answer
158 views

Dilworth's Lemma

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that either For all boxes, all pockets in a box must include a ball with the same color ...
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154 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 & ...
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55 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
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3answers
1k views

Number of equivalence relations on a finite set

I need a hint for computing the number of ways in which all the equivalent classes on a set of $n$ elements can be realized. For example, if the set has 2 elements ${a,b}$, then there are 2 possible ...
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1answer
31 views

Partitions of $\alpha$ Variation

Suppose $T>0$ . Does anyone know if there exists a sequence of partitions $(\pi_n)_{n\in\mathbb{N}}$ of the interval $[0,T]$ such that the mesh size goes to $0$, and such that it is of bounded ...
3
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1answer
55 views

Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle.

Consider perimeters $>2$ equal to $0$, $2$, or $10 \mod(12)$. The sequence starts $10, 12, 14, 22, 24, 26, 34, 36, 38, 46, 48, ...$ and we can look at the three part partitions that make triangles. ...
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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: ...
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1answer
33 views

Injection from a set to its partition

Let $A$ be a set such that $|A|=2^{\aleph_0}$ and let $S$ be a partition of $A$ such that $S$ is not countable. Is there a way to define an injective function $f:A\longrightarrow S$ in order to prove ...
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1answer
41 views

Invective function from $(0,1)$ to a partition

Consider the set $(0,1)$ and denote every $a \in (0,1)$ by it's decimal expansion $$ a=0.a_1a_2a_3\ldots $$ Now, define the equivalence relation $a \sim b$ if and only if $a_p = b_p$ for every prime ...
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0answers
20 views

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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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 $R^* = S^{-1}R$ ...
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2answers
93 views

How to use pentagonal number theorem to determine partitions of n

Under the heading Pentagonal Number Theorem > Relation With Partitions, Wikipedia gives the equation $$ p(n) = \sum_{k} (-1)^{k-1}p(n-g_k) $$ where the summation is over all nonzero integers k ...
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2answers
27 views

Proving partition function equation

Let $\pi_m(n)$ represent the number of partitions of $n$ in which no part is greater than $m$. Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n).$ I know there is a theorem that helps, but I don't ...
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2answers
119 views

Partition Identity Proof

Hey guys I am trying to prove the following identity. $p(n)\leq p(n-1)+p(n-2)$ for every ${n}\geq 1$. I worked on breaking it down into steps. I think that the best way to go about with this is in ...
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3answers
322 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 ...
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1answer
51 views

integral partition, real analysis

I'm struggling with this question: If we have $f(x) = x^2 $ and $P_n $ which partitions $[1,3]$ into $n$ sub-intervals, each equal in length,how can I write the formulas for $L(f,P_n)$ and $U(f,P_n)$ ...
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1answer
46 views

Number of possibilities in a partition problem

Given a set of n items, how many possibilities are there, to distribute these items in two sets with $\dfrac{n}{2}$ items, each? I came up with something like $\dfrac{n!}{\dfrac{n}{2}!}$ but the ...
6
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2answers
140 views

For every partition $\pi$ of a fixed integer $n$, $\sum{F(\pi)}=\sum{G(\pi)}$

I need to prove the following question. For every partition $\pi$ of a fixed integer $n$, define $F(\pi)$=number of occurrences of 1 as a summand, and $G(\pi)$=no. of distinct summands in the ...
2
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1answer
59 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 ...
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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 ...
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0answers
27 views

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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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 ...
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13 views

Prove that the number of partitions of n into 3 parts is equal to the number of partitions of 2n into 3 parts, each of size less than n. [duplicate]

Prove that the number of partitions of n into 3 parts is equal to the number of partitions of 2n into 3 parts, each of size less than n. For the partitions of n into 3 parts, we have the first row ...
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0answers
82 views

Partitioning an n-set into k same-size subsets [duplicate]

"How many ways can you partition a set of size $n$ into $k$ parts of the same size?" I've tried solving in the following way but I'm not sure if it's correct, feedback would be appreciated. I'm new ...
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2answers
49 views

Number of 1's among all partitions of an integer

I am trying find a recurrence relation for the number of 1's among all partitions of an integer. The OEIS database has an entry mentioning this particular sequence but does not give a recurrence ...
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2answers
44 views

Introduction to Analysis: Bounded and Defined

This question has been rattling my brain for a while. If $f(x)$ is bounded and defined on the interval $[a,b]$ does this imply $(f(x))^2$ is bounded and defined on $[a,b]$? I would say so. Just not ...
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0answers
53 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 ...
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1answer
158 views

Counting certain partitions of integers

[Recall that] Young's lattice is a partially ordered set in which all partitions of integers are ordered thus: The elements just one step below any partition are those that you can get by subtracting ...
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1answer
89 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
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1answer
102 views

Show that there is a sequence $(P_n)$ of partitions of $[a,b]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g(x)= \begin{cases} 0, & \text{if }x\in\mathbb{Q} \\ 1/x, & \text{if }x\not\in\mathbb{Q} \end{cases}$, $x\in[0,1]$. Show that $\exists$ sequence $(P_n)$ of tagged partitions of $[a,b]$ ...
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1answer
32 views

Partitions and Equivalence Relations [closed]

I have no idea how to solve this: Suppose that $\{S_i:i \in I\}$ is a partition of a nonempty set $Y$ and $X$ is a nonempty subset of $Y$. Prove that $\{X \cap S_i:i \in I \text{ and } X \cap S_i ...
1
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3answers
104 views

Existence of uncountable set of uncountable disjoint subsets of uncountable set

"Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?" Intuitively, I feel like it should be ...
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0answers
61 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 ...