Questions tagged [partitions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
-2
votes
0answers
60 views

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 $$x_1^2+x_2^2=d?$$...
2
votes
0answers
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 ${\...
2
votes
0answers
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 ...
4
votes
0answers
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,...
0
votes
1answer
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 ...
0
votes
0answers
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
1answer
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 ...
0
votes
0answers
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 ...
3
votes
2answers
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
1answer
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
votes
0answers
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
1answer
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 ...
7
votes
1answer
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(...
2
votes
1answer
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'...
1
vote
1answer
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\...
16
votes
3answers
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
1answer
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 ...
1
vote
2answers
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 $(...
0
votes
1answer
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 ...
8
votes
0answers
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 ...
2
votes
1answer
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 ...
1
vote
1answer
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|...
6
votes
4answers
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 ...
2
votes
2answers
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)\...
0
votes
0answers
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 ...
3
votes
1answer
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), ...
0
votes
0answers
21 views

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 ...
9
votes
2answers
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 $p(n) = p_{\{1,\ldots,n\}}(n)$....
2
votes
2answers
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 ...
2
votes
1answer
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 ...
12
votes
1answer
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 ...
3
votes
2answers
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 $...
5
votes
0answers
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 ...
7
votes
4answers
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}^...
-1
votes
1answer
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 > \...
4
votes
2answers
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?
2
votes
1answer
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 ...
5
votes
1answer
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 $\eta(k):=\sum_jk_j$. ...
3
votes
2answers
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 ...
5
votes
0answers
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, ...
4
votes
0answers
130 views

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 ...
2
votes
0answers
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{...
6
votes
1answer
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 \...
16
votes
2answers
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 $\...
0
votes
0answers
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-...
5
votes
0answers
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\...
8
votes
0answers
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 ...
5
votes
0answers
115 views

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 ...
2
votes
0answers
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(\...
1
vote
0answers
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 ...

1
2 3 4 5 6