All Questions
Tagged with integer-partitions integers
29
questions
4
votes
1
answer
2k
views
Restricted partition of integer: strictly k distinct parts from a set
I am looking for a way to compute/approach a serie of restricted partitions (or compositions) of integers.
I've found the $Q(n,k)$ quantity (OEIS) satisfying the first two of my following constraints ...
- 43
-1
votes
1
answer
33
views
Generating functions (parts equal to $2$) [closed]
What is the generating function for partitions into parts equal to $2, 5$ or $7$?
I know the function of distinct parts, but what if they don't have to be distinct?
- 3
0
votes
2
answers
175
views
Generating function for partitions into parts
My question is short and (maybe) simple: What is the generating function for partitions into distinct parts equal to $2, 5$ or $7$?
My idea is to use Euler's theorem: $$\sum p(k)x^k=\prod\frac{1}{1-...
- 185
1
vote
3
answers
2k
views
Integer partitions: number of even parts
I've got an elementary, combinatoric question: If the number n is odd, why is the number of even parts = the number of times where each part appears an even number of time = 0 ?
I mean: Of course, ...
- 185
1
vote
1
answer
906
views
Ordered Integer Partition of fixed size
I'm trying to find how many ways there are to partition n into k partitions with max value s. I've found many solutions that does this for unordered partitions. They use recursion and the say that p(n,...
- 121
2
votes
1
answer
371
views
Proof of an integer partitions inequality
I came across an interesting problem the other day.
Let $P_n$ be the number of partitions of a positive integer $n$. For instance $P_4$ = $5$, as there are five ways of partitioning $4$:
$4$
$3+1$
$...
- 233
1
vote
3
answers
128
views
Partitioning a sequence of characters into groups of $3$ or $4$
I have some computer code that outputs character sequences of various lengths. For readability, I would like to format output sequences as groups of $3$ or $4$ characters. Examples:
The sequence <...
- 163
1
vote
0
answers
115
views
Notation for "factorials" of integer partitions?
Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a partition of $n $, $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0$.
Is there accepted notation (e.g. in the literature) for ...
- 2,579
1
vote
2
answers
286
views
Simple expression for Partition function, $P(n)$, for $n\le20$
I know there are various forms to express the function $P(n)$, the number of partitions for an integer $n$, e.g. see some of the formulas referenced on Wolfram.
But say I only care about the first ~$...
- 645
6
votes
3
answers
1k
views
Integer Partition (Restricted number of parts)
I read an article about integer partition which is posted on Wikipedia, and I found out that the generating function of "partitions of n into exactly k parts" can be represented as:
$$\sum_{n\ge0} ...
- 451
0
votes
1
answer
155
views
Determine if a partition of integer using given parts exists
Is there a way to determine if a partition of integer $n$ into a given set of integers (represented by a vector $\mathbf{x}$) exists, other than enumerating all partitions of $n$ and checking if one ...
- 391
0
votes
1
answer
134
views
Proof that $ \sum_{i=1}^k p_i = (n-k) $ where $p_i (n)$ is the number of partitions of n into exactly i parts.
I have to proof that
$p_i(n) = p_i(n − i) + p_{i−1}(n − i) + . . . + p_1(n − i).$
for every $ 1 \le i \le n $, where n is number of n partitions has exactly i parts.
Then I have to calculate $p_5(...
- 123
1
vote
1
answer
3k
views
Coin Change Problem with Fixed Coins
Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<\cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins.
...
- 61
1
vote
1
answer
271
views
Count the number of ways n different-sided dice can add up to a given number
I am trying to find a way to count the number of ways n different-sided dice can add up to a given number.
For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ($(2,6),(3,5),(4,4)...
- 1,013