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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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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19 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 ...
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1answer
207 views

Two different coins on a chessboard

Two different coins are placed on squares of a standard 8x8 chessboard; they may both be placed on the same square. Let us call two arrangements of these coins on the chess board equivalent if we can ...
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2answers
351 views

Partitions of a set into three parts

How many partitions of the set $\{1,2,3, \ldots , 100\}$ are there such that both a) there are exactly three parts and b) elements $1,2,3$ are in different parts. Any help on this question would ...
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64 views

A conjecture on partitions

While trying to prove a result in group theory I came up with the following conjecture on partitions: Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for ...
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83 views

Explanation of block systems and group action

According to Wikipedia: "If $B$ is a block then $gB$ is a block for any $g$ in $G$. If $G$ acts transitively on $X$, then the set $\{gB \mid g \in G\}$ is a block system on $X$." i.e., $\{gB \mid g ...
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72 views

Monoid on ordered partitions of a natural number

Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$. We can define two ...
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1answer
26 views

Partioning Mystery

Who has the wisdom to answer the following: 9 distinct marbles distrubted into 4 distinct bags with each bag receiving at least 1 marble,how many ways can this be done? Thankyou for contributing! ...
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1answer
40 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 $S_D-s_D<2^{-1000}$.
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Partioning/Enumeration

How many ways can one distribute A) 15 Balls into 3 bags. Both bag and balls are distinct (labelled) and each bag must contain at least one ball. B) 10 balls into 3 bags. again both bag and balls ...
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33 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 ...
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41 views

Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
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1answer
58 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
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0answers
28 views

Quantiles when the population contains only one unique value

My apologies for the mixing of the terms quartile and quantile below. I am interested in the general case of quantiles, but I'm using a quartile as a specific example. Also feel free to clarify any ...
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1answer
104 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
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62 views

What is the least integer of additive dimension 4?

Say that $m$ is the additive dimension of $n\in\Bbb N$, and write $m=\operatorname{ad}n$, if $m$ is the greatest integer for which there is an irredundant $m$-element set $M\subset\Bbb N$ that ...
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2answers
49 views

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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247 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
68 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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398 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
83 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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27 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
68 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
130 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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76 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
106 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 ...
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2answers
45 views

Partitioning $\Bbb{N}$

Can we partition $\Bbb{N}$ into a finite union $$J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}$$ where $\sqcup$ denotes disjoint union. I'm guessing if we can then one of the $J_i$ must be infinite and ...
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1answer
94 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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57 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
64 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
44 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 ...
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201 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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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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1answer
34 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 ...
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62 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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43 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
50 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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1answer
36 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
31 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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163 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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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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34 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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226 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.) Question. Is ...
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153 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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1answer
71 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
57 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 ...
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61 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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42 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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29 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 ...
2
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88 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 ...