Questions tagged [partitions]

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2
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0answers
42 views

Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
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1answer
178 views

Partitioning a convex $n$-polygon

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \...
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1answer
249 views

When are immanants irreducible?

For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
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120 views

Partitions of n into k distinct parts which are multiples of given numbers

Is there anything known about the number of partitions of an integer $n$ into $k$ distinct parts in the following way? Let $a_1,\dotsc,a_k\geqslant1$ be given integers. In how many ways can we write $...
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1answer
107 views

Partity of partitions with distinct parts of parts $>1$

This question is motivated by my earlier (unanswered) MO post. The number of partitions into distinct parts is generated by $\sum_{n\geq0}Q(n)x^n=\prod_{k\geq1}(1+x^k)$. Focusing on parity of ...
2
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1answer
145 views

Objects in bijection with integer partitions (and lattices)

A partition of $n$ is a non-increasing sequence of positive integers of sum $n$. Several lattices are defined over integer partitions, in particular the dominance order and the Young lattice. Several ...
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1answer
242 views

Alternating sum of hook lengths: Part II

This is a follow up on my earlier MO post. Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Let $$f_n=\sum_{\...
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0answers
34 views

Problem concerning cutting of 2n*2n square into 2 equal area connected figures using various cuts without self crossings

We have a square 2n*2n, where n belongs to N. The main problem is to find how many different equal area connected figures could be produced by cuttings without self-crossings. The orientability of the ...
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2answers
531 views

Alternating sum of hook lengths: Part I

Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Is there a closed formula or a generating function for the ...
3
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1answer
94 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
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1answer
424 views

Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: \begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}...
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79 views

Dimension of a certain space of symmetric functions: Part II

This is the second installment of my earlier MO question. Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
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183 views

Dimension of a certain space of symmetric functions: Part I

Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$. QUESTION. Consider the ...
4
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1answer
186 views

Identity involving binomial coefficients and partitions

Working on a problem in the symmetric group I have stumbled upon the following equation: $$\sum_{\substack{\pi=(1^{c_1},2^{c_2},\ldots,n^{c_n})\\\textrm{partition of }n}}(-1)^{n-\sum_{i=1}^nc_i}\frac{...
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162 views

This sum over partitions has unexpectedly nice denominators

Fix an integer $n >= 0$, a power series $\gamma \in \mathbb Q[[X]]$ with valuation 1, and a symmetric function $f$ (with coefficients in $\mathbb Q$). Now, consider the series $$ S_n = \sum_{\...
3
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1answer
149 views

Generating function for parity in hooks

Let $\lambda\vdash n$ denote an integer partition of $n$ and $\frak{H}_{\lambda}$ be the multiset of hook lengths of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even ...
0
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1answer
113 views

Ratio limit results for restricted partition functions

This concerns difference/limit ratio results for special restricted partitions. Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...
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89 views

Writing integers as sequences of products by 2 and integer divisions by 3

For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$. For instance: $$ 100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
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1answer
202 views

Does the ordinary generating function of Bell numbers converge?

I am working in a field not really based on combinatorics, therefore I appologize if my question is in any kind invalid. Nevertheless, in my calculations, the Bell numbers appeared. I need to find ...
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1answer
214 views

A combinatorial problem about partitions [closed]

A partition of $n$ is a unordered list, whose sum is exactly $n$. The total number of $2$’s in all partitions of $n$ is equal to the total number of singletons in all partitions of $n−1$. A singleton ...
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2answers
498 views

Number of d-Calabi-Yau partitions

This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2). We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
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86 views

Lengths of cycles in non-crossing partitions

Let $g$ be an element of the permutation group $S_n$ and let $\eta$ be the cyclic permutation $(1,2,\dots,n)$. Define $D(g)$ as the number of cycles in permutation $g$. I am aware of the fact that the ...
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0answers
83 views

Integer partitions with same divisors

Definitions: For $\alpha < \beta \in (0,1)$, let $P(n,\alpha,\beta)$ be the set of unordered integer partitions of $n$ where each part of the partition has size between $n^\alpha$ and $n^\beta$. We ...
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2answers
569 views

A generalization of partition function to the sums of squares

The well known partition function $p(n)$ is defined as the number of ways to represent $n$ as the sum of natural numbers. An asymptotic formula for $p(n)$ is $$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\...
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1answer
143 views

A formula for this generating function that is similar to the $qt$-Catalan numbers

I came up with the following conjecture: $$ \sum_{n \ge 0} z^n \sum_{\mu \vdash n} \frac{ t^{\sum l}q^{\sum a}}{\prod (q^a - t^{l+1})(t^l - q^{a+1})} = \exp\left(\sum_{n \ge 1} \frac{z^n}{n(q^n-1)(t^n-...
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110 views

A conjecture about sums over partitions arising from Hilbert scheme of points

$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
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203 views

Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?

Lehmer 1988 and Keiper 1992 made major progress on evaluating the series: $$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$ where $\...
4
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2answers
236 views

Combinatorial representation of function

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0....
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0answers
101 views

A combinatorial proof of an identity of partitions (Macdonald I.5)

This is a statement from Symmetric Functions and Hall Polynomials by Macdonald: $\sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2$ where $\lambda$ denotes a partition or a Young diagram, and for each ...
3
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1answer
189 views

Under which conditions the domain of the surjective function $f:[a,b]\times[c,d]\to[0,1]^{2}$ can be split s.t. the restrictions are bijective?

This is a follow-up question to this. Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted ...
2
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1answer
266 views

Is it always possible to partition $[a,b]\times[c,d]$ into disjoint blocks $D_{ij}$ s.t. $\left.f\right|_{D_{ij}}$ is bijective?

Consider the function given by $f:[a,b]\times[c,d]\to[0,1]^{2}$ such that $0\leq a < b \leq 1$, $0 \leq c < d \leq 1$. Moreover, we do also have that $f\in C^{1}([a,b]\times[c,d],[0,1]^{2})$ and ...
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1answer
82 views

Restricted partition problem into parts with a given set of prime factors

I need a reference for the following question: Let $\mathcal{P}$ be a finite set of $k$ primes and let $f(n)$ be the number of partitions of $n$ into parts whose prime factors are restricted to the ...
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2answers
844 views

Does Rademacher's convergent series for p(n) define an analytic function?

Let $p(n)$ be the number of partitions of $n\geq 0$. We can let $n$ be any complex number in Rademacher's convergent infinite series for $p(n)$. (See e.g. equation (24) here.) For what $n$ does it ...
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0answers
91 views

bijection mapping a transversal to a transversal

The following must certainly be a standard result, so what I'm looking for is a reference, or the name of this theorem. I don't have any combinatorics books at my fingertips, but I could see this ...
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1answer
115 views

(Translation request) Hypotheses of the Blom-Fredberg bounds on denumerants?

I don't know Swedish and I'm not finding the article "G. Blom and C. E. Froberg, On money changing" translated into English... so I tried to read the original (Swedish) with the help of ...
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0answers
34 views

Integer partitions and compositions into restricted parts with a difference condition

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$ if $x_{max}-x_{min}=m'\leq m$? Looking for asymptotics that ...
4
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0answers
54 views

$3$-variable Jacobi style identity linked to generalised Frobenius partitions

I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be ...
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1answer
223 views

Integer partitions into restricted parts

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...
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0answers
71 views

Minimizing coefficients in a product related to the Rogers Ramanujan identity

Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$: $(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$... Now replace some of the ...
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0answers
90 views

Shellability and order filters in the partition lattice

Choose $n\in\mathbb N$. Let $B$ be a non-empty subset of $[n]:=\{1,2,\dots,n\}$. Consider the set of partitions of the set $[n]$ with exactly $|B|$ parts such that each part has exactly one member ...
5
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1answer
283 views

What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$?

Also asked on MSE: What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$?. Consider the set $X = \{1,2,3,\dots,n\}$. Define the collection of all $4$-subsets of $X$ by $$\mathcal A=...
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0answers
84 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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150 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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1answer
81 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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0answers
65 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 ...
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1answer
498 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 ...
3
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2answers
144 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 ...
2
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1answer
146 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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0answers
116 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 $\...
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1answer
192 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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