# Questions tagged [partitions]

The partitions tag has no usage guidance.

302
questions

**2**

votes

**0**answers

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 $|\gamma| + p w = n$. ...

**1**

vote

**1**answer

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,
$$
\...

**10**

votes

**1**answer

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_{...

**3**

votes

**0**answers

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 $...

**1**

vote

**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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_{\...

**1**

vote

**0**answers

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 ...

**11**

votes

**2**answers

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**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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}...

**1**

vote

**0**answers

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 $\...

**5**

votes

**0**answers

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**

votes

**1**answer

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{...

**4**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 $...

**0**

votes

**0**answers

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 \...

**4**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**14**

votes

**2**answers

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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**7**

votes

**2**answers

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(\...

**15**

votes

**1**answer

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-...

**5**

votes

**0**answers

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 ...

**7**

votes

**0**answers

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**

votes

**2**answers

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....

**3**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**22**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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 $...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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**

votes

**1**answer

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=...

**2**

votes

**0**answers

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 ${\...

**4**

votes

**0**answers

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,...

**0**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**-2**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 $\...

**1**

vote

**1**answer

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 ...