Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
1,470 questions
0
votes
0
answers
23
views
Cycle between ranks of partitions in a box
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 \...
- 12.3k
10
votes
1
answer
239
views
Partitioning $\{1,4,9,\cdots,500^2\}$ into subsets with equal sums
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 ...
- 101
8
votes
2
answers
760
views
Probability that four integers can be partitioned into two sets of equal sum
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 ...
- 307
1
vote
0
answers
27
views
Explicit evaluation of simple multinomial products
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 ...
- 391
0
votes
0
answers
119
views
grouping ordered partitions of an integer
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 ...
0
votes
2
answers
251
views
Conjecture on the existence of convex and concave polygons with all prime interior angles
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:
$\...
3
votes
1
answer
56
views
Looking for citation: Limit shape of a random partition, under multiple distributions
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 $\sqrt{n}$ ...
- 501
3
votes
1
answer
169
views
How to prove this sum over integer partitions is equal to the number of derangements?
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:
$...
- 964
2
votes
0
answers
38
views
How can we write the order of the centralizer of an element $g \in S_n^{\pm}$ in terms of its signed cycle type?
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^+,\...
- 21
3
votes
1
answer
179
views
Number of ways a sum can be written.
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 $...
- 355
1
vote
1
answer
81
views
Number of sums(with no same adjacent number) that can lead to a specific number
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 $...
- 73
0
votes
1
answer
77
views
Primes with Exactly One Minimal Decomposition into Smaller Primes
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:...
- 301
5
votes
1
answer
405
views
counting number of integer solutions
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 ...
- 823
0
votes
0
answers
46
views
Is there any formal phrase for the partitions with equal parts and the partitions with not every part having the same number?
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 ...
- 605
0
votes
0
answers
44
views
Explanation of sequence of partitions into four groups
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 ...
