Questions tagged [integer-partitions]

Question 1

Let $P(n, k)$ be the poset of partitions in the $k \times n$ box under diagram containment. Explicitly, $P(n, k) = \{(\lambda_1, \ldots, \lambda_k) \mid n \geq \lambda_1 \geq \cdots \geq \lambda_k \...

Question 2

Say we have the set $S = \{i^2\mid i \in \mathbb{Z}, 1\leq i \leq 500\}$. We want to partition this into 50 parts $P_1 ... P_{50},$ and the sum of the elements in all $P_i$ are equal (ie. each part ...

Question 3

Let $n\in\mathbb{N}$ and $X_1,\dots,X_4$ be independent and uniformly distributed on $\{1,\dots,n\}$. What is the probability that the tuple $(X_1,X_2,X_3,X_4)$ can be partitioned into two (multi)sets ...

Question 4

Let $N$ and $t$ be positive integers. I'm interested in multinomial products of the form $$\left(\sum_{m=0}^{N}x^m\right)^t \equiv \sum_{n=0}^{Nt}a_n x^n$$ The coefficients $a_{n}(N,t)$ can be ...

Question 5

Consider the number of ways to ordered partition an integer $x > 4$ into $x-4$ ones and $2$ twos. In each of these partitions, we want to split them into at least two groups such that the sum of ...

Question 6

Recently, I have been thinking about an interesting conjecture which I believe has never been preposed before, which I have dubbed the panprimangular polygon conjecture. The conjecture states: $\...

Question 7

I have a memory of a research article which studied the "average" shape of a randomly-chosen partition of an integer $n$, when $n \rightarrow \infty$ and scaling the partition by $\sqrt{n}$ ...

Question 8

I'm trying to figure out the value of the following sum, $S_n$. It's defined over a set $H_n$ which contains integer partitions of $n$ using only parts of size 2 or greater. Here are the definitions: $...

Question 9

Let $S_n^{\pm}$ be the signed symmetric group. It is well known that there is a one-to-one correspondence between the conjugacy classes of $S_n^{\pm}$ and ordered pairs of partitions $(\lambda^+,\...

Question 10

Let $m$ be a a positive integer. Define $\mathscr{D}_m$ to be $(m+1)(m−1)$ if $m$ is odd, and $(m+2)(m−2)$ if $m$ is even. Then let $\mathscr{N}_m$ to be $4$ or $0$ accordingly as $m$ is divisible by $...

Question 11

Let me take an example . I wanna express 6 as a sum of 4 numbers . Let we represent it as :- $$x_0+x_1+x_2+x_3=6$$ We approach this problem stepwise . First I say that there are 7 possible values of $...

Question 12

Let $p$ be a prime number. Define $n(p)$ to be the number of distinct unordered sets of smaller primes (with repetition allowed) that sum to $p$ using the minimal number of terms possible. For example:...

Question 13

Given $m,h$ we need to count the nonnegative solutions of $n_0 + n_1 + n_2 + n_3 + \dots + n_h = h$ subject to the constraint $\sum_{i = 0} in_i \equiv 0$ mod $m$. I have tried to use generating ...

Question 14

Let $n$ be a positive integer. We primarily discuss the partitions of $n$. Question: Is there any formal phrases for the partitions with equal parts and the partitions with not every part having the ...

Question 15

Can someone explain definition of sequence A340761 (https://oeis.org/A340761) from OEIS and provide example? Let's call this sequence $a(n)$. I know what integer partition (in this case I would ...

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Questions tagged [integer-partitions]

Cycle between ranks of partitions in a box

Partitioning $\{1,4,9,\cdots,500^2\}$ into subsets with equal sums

Probability that four integers can be partitioned into two sets of equal sum

Explicit evaluation of simple multinomial products

grouping ordered partitions of an integer

Conjecture on the existence of convex and concave polygons with all prime interior angles

Looking for citation: Limit shape of a random partition, under multiple distributions

How to prove this sum over integer partitions is equal to the number of derangements?

How can we write the order of the centralizer of an element $g \in S_n^{\pm}$ in terms of its signed cycle type?

Number of ways a sum can be written.

Number of sums(with no same adjacent number) that can lead to a specific number

Primes with Exactly One Minimal Decomposition into Smaller Primes

counting number of integer solutions

Is there any formal phrase for the partitions with equal parts and the partitions with not every part having the same number?

Explanation of sequence of partitions into four groups

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