The partitions tag has no wiki summary.

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### Infimum of partitions

Let $X\neq\emptyset$ be a set. A partition is a subset $P\subseteq {\cal P}(X)\setminus \{\emptyset\}$ such that $\bigcup P = X$ and any distinct members of $P$ are disjoint. We denote by ...

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### Optimal tiling for a collection of partitions

I'm interested in a possible generalization of Tiling relation on the set of partitions (the question has only been partially answered).
Let $x$ be an infinite set and let $\text{Part}(x)$ be the ...

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### Tiling relation on the set of partitions

Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of
all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$
and $t\subseteq x$. We set
$$P_{[t]} = ...

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### asymptotic for restricted partitions

Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is ...

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### For any two noncrossing partitions $p, q$ of $n$, is the graph of geodesics from $p$ to $q$ in $NC(n)$ connected?

Let $NC(n)$ denote the lattice of noncrossing partitions of $n$, and let $G$ denote the Hasse diagram of $NC(n)$ with respect to covering relations, viewed as an undirected graph.
I'm interested in ...

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### Unbalanced equipartitions

Let $K$ be a compact convex set in the plane.
Say that a perimeter-halving partition of $K$
is a partition of $K$
into two pieces by a chord (a segment with endpoints
on the boundary $\partial K$) ...

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### Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...

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### Wrong asymptotics of OEIS A000607?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...

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### Generating function of $p(25n + 24)$

In a paper titled "RAMANUJAN’S UNPUBLISHED MANUSCRIPT ON THE PARTITION AND TAU FUNCTIONS WITH PROOFS AND COMMENTARY" by Bruce C. Berndt and Ken Ono, it is mentioned that Ramanujan derived the formula ...

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### Coloring summands of given n-partition with given weights of colors

Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$
Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...

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### A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless
of the starting $n$.
For example, let $n=69$, and consider this partition:
$$
(8,8,7,7,5,5,5,5,5,4,3,3,2,2)
$$
In ...

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271 views

### Faster formula to compute sum over partitions

Let $f$ be a function from the positive integers to the real numbers (or some ring...). Let
$$(\star) \quad F(n) = \sum_{n_1 \leq \cdots \leq n_j\atop n_1 + \cdots + n_j = n} f(n_1) \cdots f(n_j), $$
...

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### Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula
$$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...

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### Sequences that represent different drawing of chords?

In combinatorics there are there are special kind of sequences, in which their terms represent the number of different ways that we can draw chords with some properties.
Actually my question is ...

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### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

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### How to prove this identity? (Perhaps related to partition) [closed]

How to prove this identity?
$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}= \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$
I will appreciate it a lot if a solution using method ...

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### Partitions whose smallest part is k

Is there a partition formula that counts the number of partitions of n whose smallest part is k ? I know there exists a smallest part formula (Andrews) but it does not answer my question.
Thank you

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### partition of an integer n into atmost k =O(log n) parts

Suppose you have a partition p of n into atmost k parts, say $$\{i_1, i_2, ..., i_j, ..., i_{k-1}\}$$
For example $\{1, 4\}$ is a partition of 10 into 3 parts (in this notation i am specifying the ...

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### An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product
...

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### What is the least integer of additive dimension 4?

Say that $m$ is the additive dimension of $n\in\Bbb N$, and write $m=\operatorname{ad}n$, if $m$ is the greatest integer for which there is an irredundant $m$-element set $M\subset\Bbb N$ that ...

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### The number of partitions between two fixed partitions

Given two partitions M and N, with $M_i \leq N_i$ for all $1\leq i\leq \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum_{\lambda: M_i\leq \lambda_i\leq N_i} ...

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### Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...

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### Length of the longest chain in dominance order

If $\Pi_n$ is the set of partitions of $n$, then for $\lambda, \mu\in \Pi_n$ we say $\mu$ dominates $\lambda$ if $\sum\limits_{i=1}^k \lambda_i \leq \sum\limits_{i=1}^k \mu_i$ for all $k$. This gives ...

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### Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...

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### Number of representations of an integer as an (arbitrary) sum of products

If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...

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### Generating random weak k-bounded reverse plane partitions

Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that ...

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### Asymptotics of partitions in at most n parts, bounded by r

I posted this question on MathStackexchange (http://math.stackexchange.com/questions/639878/asymptotics-of-partitions-in-at-most-n-parts-bounded-by-r) some time ago, but it did not receive any answer, ...

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### Combinatorial Technique Needed

The following problem is likely too special for MO.
However I have no clue how to deal with it, so I'll just try. Nevertheless
it is a combinatorial problem and a discussion about general methods
in ...

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### Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset S of the xy-plane is star-convex if there is ...

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### Terminology for a Partition of a Set which Includes Empty Sets

Mostly I see a partition of a set A defined as a collection of non-empty disjoint sets whose union is A.
I see one reference that allows empty sets to be included in the partition: ("Potter, M. "Set ...

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### Computing the lexicographic indices of integer partition

If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...

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### Number of 3-tuple partitions of a multiple of three which follow the triangle inequality

Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ .
If I chose any one partition randomly from ...

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### Running the Greene-Nijenhuis Algorithm Backwards

This question is crossposted from math.stackexchange.com, where it remains unanswered.
Let $Y$ be a Young tableau of shape $\lambda:=(\lambda_1,\ldots,\lambda_n)$, where ...

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### A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le ...

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### Recurrence relation for coefficients of product of generating functions for partition numbers

It is well known that
$$Z(x,q) = \prod_{n=1}^\infty\frac{1}{(1-xq^n)} = \sum_{m=0}^\infty\sum_{k=0}^m p_{m,k}x^kq^m$$
is the generating function for the number $p_{m,k}$ of partitions of $m$ in ...

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### Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...

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### Between Cover and Partition

In a cover problem, there is a complex shape (e.g. a polygon), and we have to find a set of simpler shapes (e.g. squares or rectangles), such that their union is exactly equal to the complex shape.
A ...

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### Does the asymptotic formula for Partitions into parts <c exist?

A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda_1,\lambda_2,\lambda_3,....\lambda_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands ...

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### Databases for sequences indexed by partitions

Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...

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### bounded partitions and bounded signed partitions of integers

Define a bounded signed partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation:
$$n = \pm a_{1} \pm a_{2} \pm a_{3} \pm \dots \pm a_{m}$$ where each $a_{i}$ is a integer in ...

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### Algorithm for load balancing during exchange between pairs

There are $P$ buckets each with $n_p (p \in 1..P)$ items. Let $a$ be the average number of items, so $a = \frac{\sum_{p=1}^P n_p}{P}$.
Now for each pair of buckets $i$ and $j$, we need to transfer ...

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### Rate of Convergence for Limit Shape with Integer Partitions

There is a known phenomenon in integer partition theory that almost all integer partitions, after a normalization ($\pi/\sqrt{6n}$ where $n$ is the norm of the partition), have young diagrams which ...

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### Random RSK and Plancherel Measure

Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...

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### Why are the dinv-statistic and the partition length equidistributed?

A partition of $n$ is a weakly decreasing sequence of natural numbers $\lambda = (\lambda_1, \lambda_2, \dots)$ such that $\sum \lambda_i = n$. Its length $l(\lambda)$ is the number of positive ...

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### Distribution of colors in the number of integer partitions of n

Given an integer $n$ the number of partitions of $n$ into two colors can be represented as
$$p_2(n)=\sum_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the ...

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### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...

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### What can be proved about the Ramanujan conjecture using elementary means?

The Ramanujan conjecture states that the coefficients $\tau(n)$ in the identity
$$q\prod_{m=1}^\infty(1-q^m)^{24}=\sum_{n=1}^\infty\tau(n)q^n$$
satisfy the inequality $|\tau(n)|\leq d(n)n^{11/2}$, ...

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### 3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations
of permutations, RSK correspondence, contingency tables,
etc. One reason I am interested in this is because it is potentially
related ...

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### Maximal Zero Sums Partition

You are given $n$ numbers between $-n$ and $n$, the sum of numbers is $0$. Divide the given sequence on disjoint subsequences in such a way that each subsequence has zero sum. Each element should ...

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### algorithmic almost equitable partitioning

Let $G$ be a graph -- possibly infinite, but I will be glad to learn a positive result even in the finite case. Then the trivial partition (i.e., one cell coinciding with the whole $G$) is clearly ...