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Integer partitions under divisibility constraint

Consider integer partitions of $x \in \mathbb{N}$ of size $k$ under the constraint that the partition elements are distinct and the ratio of any element to each smaller element is a natural number. ...
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Equitable partition

This is in reference to this question: equitable partitions Suppose I have this graph enter image description here whose equitable partition can be taken as $\{1,3,5,7\} ;\{6,2,4,8\}$ But then as ...
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Number partitions

(I'm trying to solve a problem for computer programming. Don't have much of a math background, so I hope I am using the right terminology) Is there a formula for getting the partitions of a number ...
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197 views

Generating function for 3 -core partitions: Part II

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$. We call $\lambda$ a $t$-core partition if none of ...
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Generating function for $3$-core partitions

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$. We call $\lambda$ a $t$-core partition if none of ...
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A link between hooks, contents and parts of a partition

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Denote its conjugate partition by $\lambda'$. For example, if $\lambda=(4,3,1)$ then $\lambda'=(3,2,2,1)$. ...
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SYT and contents of a partition

Let $\lambda$ be an integer partition, denote the number of Standard Young Tableaux of shape $\lambda$ by $f_{\lambda}$. This number is computed by the formula $$f_{\lambda}=\frac{n!}{\prod_{u\in\...
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Partitions and $q$-integers

Denote an integer partition of $n$ by $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)$ where $\lambda_k>0$. Also recall the $q$-analogues of integer $n$ given by $[n]_q=\frac{1-q^n}{1-q}$. ...
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Partitions, $q$-polynomials and generating functions

Recall the integer partition function $P(n)$ with generating function $$\sum_{n\geq0}P(n)x^n=\prod_{k=1}^{\infty}\frac1{1-x^k}.$$ Let $[n]_q=\frac{1-q^n}{1-q}$ denote the $q$-analogue of the integer $...
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Collapsed partitions and generating functions

Given $n\in\Bbb{N}$, the number of (unrestricted) integer partitions of $n$ are given by $$\sum_{n\geq0}p(n)x^n=\prod_{j\geq1}\frac1{1-x^j}.$$ Define the collapsed partitions of $n$ to be the ...
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Partition of 4-tuples

Some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ are given, with all $a_i,b_i,c_i,d_i\leq 1$. Can we always partition $\{1,2,\dots,n\}$ into two subsets $X,Y$ so ...
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Partitioning ordered objects into sets so as to maximize isolated objects

I'll phrase this as how I thought of it: Henri is a headmaster who is planning a school trip. He has ordered $n$ coaches to take his $m$ students to the trip destination, and the students have ...
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1answer
116 views

Integer partitions with subset sums “not divisible” by p

I have the following questions: Let $N \in \mathbb{N}$ and \begin{equation} \sum_{i=1}^k n_i = N, \end{equation} with $n_i \in \mathbb{N}$ for $1 \le i \le k$ and some $k \in \mathbb{N}$, be an ...
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Maximizing the sum of hook lengths

Given positive integers $a\geq b$, and $n\in\{1,2,\dots,ab\}$ I am looking for a partition of $n$ into at most $b$ parts of size at most $a$ which maximizes the sum of the hook lengths in the ...
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sets of partitions associating any two elements exactly once

There may be a theory that deals with problems like this but I'm not enough of a mathematician to know what it is. So far I've looked up braid groups, block design, and the recommended related posts ...
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Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$

Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since $$9 = 1^2 + 1^2 + 1^2 + ...
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Distance to edge of voronoi diagram

I have $N$ points in an inner product space $S$ with inner product $<\cdot,\cdot>$. Construct a Voronoi partition, $V$, of $S$ with those $N$ points, using the metric induced by the $L_2$ norm. ...
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Congruences Ramanujan-style

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating ...
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222 views

Hamiltonian cycles — Partition function

Assume a complete undirected graph $G'=(\mathcal{V}',\mathcal{E}')$ and the partirion function: $$\sum_{\boldsymbol{x}\in \{-1,+1\}^n} \prod_{\left(i,j\right)\in \mathcal{E}'} \left[1+x_{i}x_{j}\...
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vector partition

I am interested in partitioning a vector with nonnegative integer entries into a sum of vectors with nonnegative integral entries. For example the partitions like (2,2) = (1,1)+(1,1) = (2,0)+(0,2) = (...
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Relation between a continued fraction and partitions

I am interested in the continued fraction $$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$ OEIS A104977 states ...
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Wherefore art thou a Borcherds Product?

This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
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457 views

Which of these sums appear most often?

Let $N=\{1,2,3,\ldots, n\}$. We sum all the elements of every nonempty subset of $N$. Which sum(s) appears most often? (Let's call this sum a champion). Using a simple pigeonhole argument a champion ...
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111 views

Partitions of finite sets and their behavior under permutations of the set

The following seems to be useful, and probably well-known, but I can't find a reference for it. If anyone can point me to a textbook or paper which states it, then I'd be grateful. Consider a ...
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A property of 47 with respect to partitions into five parts

Is 47 the largest number which has a unique partition into five parts (15, 10, 10, 6, 6), no two of which are relatively prime?
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Generalized partitions and eta functions

Let $\sigma$ be an element of $SL_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the ...
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1answer
135 views

Yet another question about unrestricted partitions

I posed a question called "A Product Related to Unrestricted Partitions". As it stands it is too hard. Here's another variation which is easier to search for and hopefully might shed some light on ...
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Upper bound for number of subpartitions of a partition

Let $\alpha = (\alpha_1,\alpha_2,\dots)$ be a partition of the positive integer $n$, that is, a nonincreasing sequence of nonnegative integers $\alpha_j$, with only finitely many nonzero terms, whose ...
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Sequences of nested integer partitions

I'm looking for whether the following objects have a name or have been studied at all in the literature: sequences of partitions $(\lambda^{(1)},\lambda^{(2)},\ldots)$ which are nested, i.e. $\lambda^{...
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Another characteristic of subsets of (finite) power sets

Consider a finite set $M$, its power set $\mathcal{P}(M)$ and a subset $S$ of the power set, i.e. $S \subset \mathcal{P}(M)$. Consider as a characteristic of $S$ the function $f_S: \mathbb{N} \...
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The number of permutations of a given cycle type that fix a string with a given histogram

Let $\lambda$ and $\mu$ be partitions of some integer $n \geq 1$. Let $d$ be the number of parts in $\mu$ and let $\bar{\mu} \in \{1,\dotsc,d\}^n$ denote the string $1^{\mu_1} 2^{\mu_2} \dotsb d^{\...
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Generating function of prime power partitions

By a theorem of Luck, $K^0 (BS_n) \simeq \mathbb{Z} \times \prod_p (\mathbb{Z}_{p}^\wedge)^{r(p,n)}$ where $r(p,n)$ is the number of partitions of $n$ into powers of $p$ (excluding the trivial ...
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118 views

Partitions of the reals such that closures of partition elements are saturated

Suppose we have a partition $P$ of a set $S$ and a unary set operation $u:\mathscr{P}(S)\to\mathscr{P}(S)$ such that for each $A\in P$ the set $uA$ is saturated with respect to $P$ (a union of ...
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1answer
68 views

Existence of a set partition satisfying some restriction

I am looking in the literature for references to combinatorial result of the kind of the one below. I am quite sure they (or some variations of them) should have been studied intensively, but now I am ...
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77 views

Source coding lexicographic index of finite alphabet sequence with weight (partitions)

My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint: $$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
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What is the growth rate of the number of unoriented cobordism classes?

Let $\Omega_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega_n^O\rvert$. What are the asymptotics of $d(n)$? It's ...
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A functional on paths in a symplectic vector space

I'm running into a functional associated to a piecewise smooth curve $\gamma: [0,1] \to V$, where $V$ is a real vector space with a symplectic form $\omega$: $$ \int_{0 \leq x \leq y \leq 1} \omega(\...
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Wilf's conjecture: complementary Bell numbers

The complementary Bell numbers or Uppuluri-Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by $$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$ Definition. Fix an integer $m\geq0$....
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215 views

A discrete operator begets even/odd polynomials

Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$. Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
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Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?

In this paper, the following result is proved. For any prime $p$, all the Fourier coefficients of $$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$ ...
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What is the relationship between Partition function and Betti numbers for nilpotent Lie algebras?

Let $p(n)$ denote the number of partitions of a positive integer $n$. It is known that $\{p(n)\}_{n>25}$ is log-concave. Dietrich Burde said in this MathOF post that property $PF_3$ for partition ...
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“strange” diophantine and parity of the partition function

Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$ be the sequence of those positive integers of the form $$ p^{4\alpha+1}n^2$$ in increasing order where $p\equiv 5\pmod 8$ is prime ...
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Equi-distribution of the parity of partitions

The integer partition function $p(n)$ has a generating function given by $$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$ with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
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Hooks in a rectangle: Part II

This problem is a follow up on my other MO question. On the basis of experimental data, I'm prompted to ask: Question. Let $R(a,b)$ an $a\times b$ rectangular grid, $h_{\square}$ the hook-length ...
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Hooks in a staircase partition: Part I

This quest has its impetus in a paper by Stanley and Zanello. I became curious about What is the sum of all hooks lengths of all partitions that fit inside the $n$-th staircase partition? On the ...
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Proportion of partitions in a rectangle

Let $ k, n, r \geqslant 1 $ be integers. Let $ \lambda $ be a partition of $r$, what we denote by $ \lambda \vdash r $. I would like a lower and an upper bound for the following quantity, for all $ ...
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194 views

Counting Bipartitions

Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$. Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
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A thickening map on integer partitions

I am looking for the name of the following map $t(\mu)$, defined for integer partitions $\mu=\mu_1\geq\mu_2\geq\dots$: if $\mu$ is empty, return $\mu$. if the first part $\mu_1$ of $\mu$ is at least ...
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Max+Min of a specific data set

Let $r\geq 6$ and $3\leq f \leq 2r-3$ be integers. Suppose we have $f$ data points $x_0,x_1,\ldots, x_{f-1}$ such that for each i, $1\leq x_i \leq r-4$ is an positive integer. for each i in $\mathbb ...
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390 views

In search of a combinatorial reasoning for a vanishing sum

Assume $s, j \in\mathbb{N}$. Define the set $$\mathcal{A}_{j,s}:=\{(n_1,n_2,\dots,n_j)\in\mathbb{Z}_{\geq0}^j\vert \, n_1+2n_2+\cdots+jn_j=j, \, n_1+n_2+\cdots+n_j=s\}.$$ Question. Is there a ...