# Questions tagged [partitions]

The partitions tag has no usage guidance.

267
questions

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### Number of distinct sum of two squares under sum condition [closed]

Given integer $0\leq n$ and $x_1+x_2=n$ holds with $x_1,x_2$ non-negative what is the
distribution of
and maximum
number of distinct $d$'s achievable in the sum of squares function $$x_1^2+x_2^2=d?$$...

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74 views

### Generalized partitions

Let $\kappa>0$ be a cardinal and $X$ be a set. We set $[X]^\kappa = \{A \in {\cal P}(X): |A| = \kappa\}$.
If ${\cal A}\subseteq {\cal P}(X)$ we say that ${\cal B} \subseteq {\cal P}(X)$ is an ${\...

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51 views

### Lebesgue measure of set of equidistant points with respect to a finite set

Let $X$ be a finite subset of $\mathbb{R}^n$; equip $d$ with a metric, and let $\emptyset \subset X\subseteq \mathbb{R}^n$ be of cardinality $N>0$.
What requirements on my metric do I need so ...

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143 views

### Does $\mathbb{R}$ have a partite subbase?

If $X\neq \varnothing$ is a set we say that ${\frak P} \subseteq {\cal P}(X)$ is a partition of $X$ if
$\bigcup{\frak P} = X$, and
$P\neq Q \in {\frak P} \implies P\cap Q = \varnothing$.
Let $H = (V,...

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votes

**1**answer

67 views

### Optimal partition search

Given an integer $n$, and 2 real sequences $\{a_1, \dots, a_n\}$ and $\{b_1, \dots, b_n\}$, with $a_i$, $b_i$ > 0, for all $i$. For a fixed $m < n$ let $\{P_1, \dots, P_m\}$ be a partition of the ...

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57 views

### Write large $n$ as $n_1+\ldots+n_k\ (n_1<\ldots<n_k)$ with $\varphi(n_1),\ldots,\varphi(n_k)\in\{x^k:\ x\in\mathbb Z\}$

Let $\varphi$ denote Euler's totient function.
QUESTION. Is it true that for each positive integer $k$ large integers $n$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive ...

**-2**

votes

**1**answer

424 views

### Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now
Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...

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29 views

### How many partitions are there regarding Tverberg's theorem?

"Specifically, for any set of $(d+1)(r-1)+1 $ points there exists a point $x$ (not necessarily one of the given points) and a partition of the given points into $r$ subsets, such that $x$ belongs to ...

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votes

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114 views

### Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function
$$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$
I am interested on getting ...

**1**

vote

**1**answer

64 views

### Asymptotic for restricted compositions into k parts

For every set of natural numbers $A$ and for all positive integers $n$, $k$, let $c_k^A(n)$ be the number of compositions of $n$ into $k$ parts from $A$, that is, the number of $(a_1, \dots, a_k) \in ...

**2**

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104 views

### Coefficents of these partition-based polyomials are $0, \pm1$

This is a follow up on my earlier MO question.
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate $\...

**1**

vote

**1**answer

185 views

### Divisibility of polynomials over partitions

This is a follow up from my earlier MO question.
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its ...

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votes

**1**answer

452 views

### Sum of squares and partitions

This is an off-shot from my previous post on MO.
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$, denote $\ell(\lambda)$ to be the length of $\lambda$.
Let $r_2(...

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votes

**1**answer

362 views

### An identity for polynomials over partitions

Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its conjugate partition $\lambda'$. Denote by $\lambda'...

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vote

**1**answer

407 views

### What is the closed form of this function?

It is well-known that the binomial coefficient has some monotonicity,
and that can be used to find the maximum and minimum of binomial coefficients.
Similarly, now let $\left(\delta_{i,j}\right)_{6\...

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**3**answers

604 views

### Plane partitions with equal margins

A plane partition of $n$ is an table of integers $A=(a_{ij})$ which add up to $n$ and non-increase in rows and columns. For example,
$$A= \begin{matrix} 331 \\
32 \ \ \\
11 \ \
\end{matrix}
$$
is a ...

**2**

votes

**1**answer

72 views

### 2-quotient of integer partition

This question is mostly about understanding the notation used in the following article:
Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry ...

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vote

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147 views

### Recurrence relation for the number of partitions of an integer 𝑛 with distinct summands

Let $Q(n)$ give the number of ways of writing the integer $n$ as a sum of positive integers without regard to order with the constraint that all integers in a given partition are distinct. Equation $(...

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**1**answer

124 views

### Size of parities in counting partitions into odd parts

Let $p_{odd}(n)$ be the number of partitions of $n$ into odd parts (see here). For instance, one has the generating function
$$\prod_{k\geq1}\frac1{1-q^{2k-1}}.$$
QUESTION. What is the size of this ...

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262 views

### Possible oversight in paper of Greene and Kleitman on chains in dominance order on partitions?

This question is about a possible lacuna in a paper of Greene and Kleitman which Zarathustra Brady made me aware of.
The paper in question is "Longest Chains in the Lattice of Integer Partitions ...

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**1**answer

220 views

### Asymptotics of the Steenrod algebra / $s$-partitions?

Recall that an $s$-partition is a partition of a natural number $n$ such that each part is of the form $2^r-1$. By a fundamental theorem of Milnor, the number $p_s(n)$ of $s$-partitions of $n$ counts ...

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**1**answer

139 views

### Finding Littlewood-Richardson coefficients without using identities

The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...

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297 views

### Unique partitions of two numbers

We say that $A=\sum_{i=1}^a a_i$ and $B=\sum_{j=1}^b b_j$ is a unique partition of $A$ and $B$ if there is no other way to partition the $a+b$ numbers into two parts that sum to $A$ and $B$. This is ...

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192 views

### Compositions $(n_1,…,n_r)$ of an integer $m$ such that $i$ divides $n_i$

I am studying the compositions $(n_1,...,n_r)$ of an integer $m$ such that $i\vert n_i$ for all $i=1,...,r$. (Recall that a composition $(n_1,...,n_r)\vDash m$ of $m$ is just a sequence $(n_1,...,n_r)\...

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116 views

### Is there exists a lattice isomorphism?

Let $\text{P}$ be the set of partitions of {1,2,...,n} and $\text{Y}$ the set of Young subsets of permutation group S(n)(the coxeter group of type An).
As is well-known, the set $\text{Y}$ is a ...

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130 views

### Strict unimodality of bipartite partitions

For non-negative integers $k$ and $l$ let $p(k,l)$ denote the number of vector partitions of $(k,l)$. In other words, $p(k,l)$ is the number of ways of writing
$$
(k,l) = (k_1,l_1)+\dotsb + (k_r,l_r),
...

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### Iterated Inverse structures: polynomial representation of integer partitioning of preimages in Sigma Matrices (reference request)

I am studying iterated preimage structures of functions on a finite set.
The main structure of interest to me, the Sigma Matrix, is derived from a matrix listing the element-wise preimage sets at ...

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271 views

### Every possible number of partitions by restricting parts?

Write $p(n)$ for the number of integer partitions of $n$. For $S \subseteq \{1, \ldots, n\}$, let $p_S(n)$ be the number of partitions of $n$ with all parts in $S$. So $p(n) = p_{\{1,\ldots,n\}}(n)$....

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125 views

### enumerate line partitions of points in the plane

Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...

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**1**answer

83 views

### Given a set of $n$ points in $[0,1)^d$, how do I partition the space into hyperrectangles such that each hyperrectangle contains exactly one point?

I'm new to this forum so I apologize if my question is ill-posed or too general. I have the following problem. Given a set of $n$ points in the unit hypercube, $[0,1)^d$, how can I partition the unit ...

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461 views

### Dividing a chocolate bar into any proportions

Suppose I have a chocolate bar of integer length $L$, and there are $m\leq L$ people that are going to share it. We do not know ahead of time how much each person should receive, all we know is that ...

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181 views

### Algorithm to list all Kostant partitions

Let $\Phi_+$ be the set of positive roots in some root system, and let $Q_+$ be the positive part of the root lattice, i.e., the set of elements of the form $\sum_{\beta\in \Phi_+}m_\beta\beta$ with $...

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105 views

### Complementary Bell numbers $B^{\pm}(24n+14)$

The complementary Bell numbers $B^{\pm}(n)$ are defined by the alternating sum of the Stirling numbers of the second kind, $S(n,k)$:
$$B^{\pm}(n)=\sum_{k=0}^n(-1)^kS(n,k),$$ and they count the ...

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votes

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399 views

### Maximum conjugacy class size in $S_n$ with fixed number of cycles

Context: It is well known that given a permutation in $S_n$ with $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by
$$ \frac{n!}{\prod_{j=1}^...

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**1**answer

115 views

### Collapsed partitions and ordinary partitions

Adopt the standard notation for integer partitions, writing $\lambda_1^{a_1} \cdots \lambda_k^{a_k}$ as shorthand for the partition $a_1 \lambda_1 + \cdots + a_k \lambda_k$ with parts $\lambda_1 > \...

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281 views

### Number of integers for which $np(n)$ is a perfect square

Let $p(n)$ be the partition function. Are $n=1,2,3$ the only cases for which $np(n)$ is a perfect square?

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**1**answer

117 views

### Reading off top hook-lengths in partitions

Given an integer partition $\lambda$ and its Young diagram $Y_{\lambda}$, let $h_{\lambda}(i,j)$ stand for the corresponding hook length of the cell $(i,j)\in Y_{\lambda}$. Write $\lambda\vdash n$ for ...

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**1**answer

364 views

### A binary hook-length formula?

This is purely exploratory and inspired by curiosity.
Setup: For an integer $k>0$, let $k=\sum_{j\geq0}k_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum_jk_j$. ...

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158 views

### Families of ordered set partitions with disjoint blocks

Let $C_1,\dots, C_m$ be a family of ordered set partitions of $[n]$ with exactly $k$ blocks.
Write $C_i = \{B_{i1}, \dots, B_{ik}\}$ for $i=1,\dots, m$ where $B_{ij}$ are the blocks of the ordered ...

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85 views

### Hooks, monomers, dimers and Young diagrams: Part II

As promised, I've upgraded my last question.
Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...

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### Hooks, monomers, dimers and Young diagrams: Part I

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it.
Consider the one-line partition $\lambda_n=(n)$ and its ...

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125 views

### Newman's conjecture of Partition function

(Sorry for my poor english....)
Let $p(n)$ be a partition function and $M$ be an integer. Newman conjectured that for each $0\leq r\leq M-1$, there are infinitely many integers $n$ such that
\begin{...

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296 views

### hook-length formula: “Fibonaccized”: Part II

This is a natural follow-up to my previous MO question, which I share with Brian Hopkins.
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...

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1k views

### hook-length formula: “Fibonaccized” Part I

Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...

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60 views

### Is the fundamental partition associated to $n$ the partition of $r_{0}(n)$ in $k_{0}(n)$ parts that maximizes entropy?

As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-...

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341 views

### $\text{Determinant}=(\sum \text{Determinant})^2$

Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...

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216 views

### Formula for number of edges in Hasse diagram of Young's lattice interval

There is a determinantal formula for the number of elements of the interval $[\mu,\lambda]$ of Young's lattice between two partitions due to Kreweras and MacMahon in the case of $\mu=\varnothing$ (see ...

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### Identity for classes of plane partitions

There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...

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84 views

### How to prove this identity on summations and partitions?

Let $f$ be a symmetric function of $s$ variables. The identity is
$$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\...

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195 views

### Parity of number of partitions of $n!/6$ and $n!/2$

The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition ...