Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integer-partitions]

Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.

In number theory and combinatorics, a partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers. Two sums that differ only in the order of their summands (also called parts) are considered the same partition. For example, all of the partitions of $4$ are $1 + 1 + 1 + 1$, $2 + 1 + 1$, $2 + 2$, $3 + 1$, or $4$.

The number of partitions of $n$ is given by the partition function $p(n)$. For the example above, $p(4) = 5$.

Partitions can be visualized graphically with Ferrers diagrams.

Partitions have applications in symmetric polynomials, the symmetric group, and group representations.

1405 questions
1
vote
2 answers

Partitions of sums of $k$ random odd/even $r$th powers from array of consecutive $r$th powers

Like my previous question, I'll pose this one too with an array. $1^r, 3^r, 5^r, 7^r, 9^r$ (all odd $r$th powers) That's array 1. And array 2; $2^r, 4^r, 6^r, 8^r, 10^r$ (all even $r$th powers) Let's take the sum of random $k$ integers from each…
Mach9
  • 135
1
vote
0 answers

Given $x = \sum_{i \in k} a_i$ from $\{ a_1, \ldots, a_n \}$, partition $x$ as $\sum_{j \in k} a_j$

Let me put this into an example. We have an array of the first $r$ (example $r=7$) natural numbers; $1, 2, 3, 4, 5, 6, 7$ and given $k=3$, so you get to select any three integers from the pool above. Taken $1$, $6$ and $7$, we have their sum $14$.…
Mach9
  • 135
1
vote
1 answer

Prove that A001399 == A069905

While solving PE and reading SICP I found, that there are two problems, that produce the same OEIS sequence: http://oeis.org/A001399 is a number of partitions of n into at most 3 parts http://oeis.org/A069905 is a number of partitions of n into 3…
Nakilon
  • 181
1
vote
0 answers

What is wrong with this "inference" about partitions?

Given that: $p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+⋯$ Why can one not state: $p(n)≥p(n-1)+p(n-2)-p(n-5)-p(n-7)$ Here is the logic: the subsequent 2 terms of the relation are additive and the 2 following one are negative, etc. yet the magnitude of p(k)…
Just_a_fool
  • 2,256
0
votes
1 answer

On a certain kind of integer partitions

Background: I am investigating partitioning integers using a fixed radix base. Problem: Suppose we have $x$, an arbitrary integer. Let $M = \{2, 3, 5, ..., p\}$ be a set of prime moduli. The prime factorization $$x = 2^{e_0}3^{e_1}5^{e_2} \cdots…
vvg
  • 3,311
0
votes
1 answer

How many Young diagrams with length of at most $p$ rows and $q$ columns

How many Young diagrams are there with length of at most $p$ rows and $q$ columns, without restrictions on the weight of the diagrams? The previous 2 questions asked for Total number of Young diagrams with weight 6 and Total number of Young…
John Doe
  • 502
0
votes
0 answers

The binary partition function satisfies congruence $b(2n) \equiv 1+a(n) \pmod 4$ for all $n \ge 1$.

Let $b(n)$ be the binary partition function of $n$, i.e., the number of partition of $n$ for which all parts are the powers of two, and have properties $b(2n+1)=b(2n)$ and $b(2n)=b(n)+b(2n-2)$, for all positive integer $n$, where $b(0)=1$. Let…
user1089451
0
votes
0 answers

Distribution of integer partitions by number of parts

I imagine someone has studied this already so I'm just looking for resources: what sort of asymptotic behavior does the distribution of integer partitions of $n$ counted by the number of parts exhibit? Looking at the plots that WolframAlpha supplies…
failedentertainment
  • 804
0
votes
0 answers

Number of partitions of a positive integer with largest part not exceeding k

What is the number of partitions of $n\in \mathbb{N}$ so that the largest part is at most $k$? I tried it for a few small cases, but I am unable to find a general way.
user948104
0
votes
1 answer

$l$-regular bipartition

I know about $l$-regular partition. Now I came across $l$-regular bipartition/tripartition. But how are former and latter different. Where former ensures that each part in the partition is not divisible by $l$, if the latter does the same, how are…
user511110
  • 312
0
votes
2 answers

Question on combinatorics, partitions.

Let $p$ ($n|$distinct odd parts) be the number of partitions of $n$ into distinct odd parts. Prove that $p(n)$ is odd if and only if $p$($n|$distinct odd parts) is odd by using the theorem on self-conjugate partitions. Ok, so we count the hooks in…
user67253
0
votes
1 answer

Partition functions

i'm looking at this question. "$n$ is positive integer, $T(n)$ is the number of the partitions with odd parts of $n$ integer. Show that $T(n) ≡ 0 \text{ (mod } 2)$ if $n \neq \dfrac{k(3k ∓ 1)}{2}.$" i know about this question that the numbers of…
user759578
0
votes
1 answer

What is this function

Let $\left(\delta_{i,j}\right)_{5\times5}$ be a matrix with $\delta_{i,i}=1$ and $\delta_{i,j}=\delta_{j,i}=1$ or $0$ when $i\ne j$ and $\delta_{i}=\sum_{j=1}^{5}\delta_{i,j}$. Let…
EmmaT
  • 1
0
votes
0 answers

Find all the self-conjugate partitions

Find all the self-conjugate partitions of 2018, whose second part is 3. Find all the self-conjugate partitions of 2019, whose second part is 3. I'm not sure how to approach this problem.
user734715
  • 21
0
votes
1 answer

Alghoritm for getting the partitions of n into power of m?

So the problem here is to partition an integer $n$ in terms of power of $m$. That is, if we had $n = 9$ and $m = 3$ the partitions would be: $$ \begin{split} 9&\\ 3&+3+3\\ 3&+3+1+1+1\\ 3&+1+1+1+1+1+1\\ …
Darius Larpos
  • 3
1 2 3
4
5 6