All Questions
Tagged with integer-partitions summation
36
questions
3
votes
2
answers
114
views
high school math: summands
Let's say we have a question that asks you to find the amount of all possible integers adding up to a random number, lets just say 1287. However, the possible integers is restricted to explicitly 1's ...
- 78.3k
19
votes
3
answers
101k
views
Number of possible combinations of x numbers that sum to y
I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.
An asymptotic ...
- 123
0
votes
0
answers
28
views
A sum of multinomial coefficients over partitions of integer
I denote a partition of an integer $n$ by $\vec i = (i_1, i_2, \ldots)$ (with $i_1, i_2, \ldots \in \mathbb N$) and define it by
$$
\sum_{p\geq1} p i_p = n.
$$
I set
$$
|\vec i| = \sum_{p\geq1} i_p.
$$...
- 2,236
0
votes
1
answer
44
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Find closed form of real valued function
Let $f:\mathbb{N}\rightarrow\mathbb{R}, h:\mathbb{N}\rightarrow\mathbb{R}$ be two functions satisfying $f(0)=h(0)=1$ and:
$$f(n) = \sum_{i=0}^{n}h(i)h(n-i)$$
Find a closed form for $h(n)$ in terms of $...
- 14.7k
1
vote
0
answers
36
views
Partition of n into k parts with at most m
I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate
$$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$
My approach was ...
- 29
0
votes
1
answer
76
views
Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter.
I am interested in the following two quantity
(1) $$...
- 173
2
votes
1
answer
118
views
Number of partitions of $n$ into distinct even parts.
Question from my last exam:
Let $r_n$ denote the number of partitions of $n$ into distinct parts. Prove that
$$
\sum_{i=0}^n(-1)^i r_i r_{n-i}
$$
is the number of partitions of $n$ into distinct even ...
- 109k
0
votes
2
answers
40
views
max length of list s.t sum of the powers list equals n
I am trying to find out what is the best upper bound on the size of a list such that
All its values are integers between $1$ and $n$
Its values are all different from each other
The sum of the $k^\...
- 78.3k
1
vote
0
answers
89
views
A summation formula for number of ways $n$ identical objects can be put in $m$ identical bins
A famous counting problem is to calculate the number of ways $n$ identical objects can be put into $m$ identical bins. I know that this problem is somewhat equivalent to Partition problem. There is no ...
- 47.4k
2
votes
1
answer
125
views
Is there a pattern to the number of unique ways to sum to a number?
I don’t think there is a proper name for these so I will refer to them as “phactors”. Basically, a phactor is a way to sum up to a number using positive real integers that are non zero and not equal ...
- 6,574
0
votes
0
answers
61
views
Compute a certain sum [duplicate]
How would I find a formula for $$S(n,r) = \sum_{i_1+\ldots+i_r = n,~(i_k)\in\mathbb{N}^r} ~~~i_1\ldots i_r ~~~~.$$
It's easy to find that it satisfies $$ S(n,r+1) = \sum_{j=0}^n(n-j)S(j,r),$$ which ...
- 181
0
votes
1
answer
63
views
Summing over tuples $(b_1, \ldots, b_n)$ with $\sum ib_i = n$ and $b_j = k$.
Let $T_n$ denote the set of $n$-tuples $\left(b_1, \ldots, b_n \right)$ of non-negative integers such that $\sum_{i=1}ib_i=n.$ I am trying to simplify the sum
\begin{align*}
\sum_{\underset{b_{j}=k}{\...
- 3,918
1
vote
1
answer
61
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How many ways to get a sum of 29 by adding 5 & 2? ex 5+5+5+5+5+2+2 = 29, is one way.
Ex: $2+2+3$, $2+3+2$, $3+2+2$ these are three ways to get a sum of $7$ with $3$ and $2$. But my example is with $5$ and $2$ and a sum of $29$.
I believe there are three ways to get a sum of $29$ by ...
- 19.6k
1
vote
1
answer
114
views
summation of product of binomials coefficients over compositions
I am having trouble with this problem which arises in the context of computing lowest theoretically possible computation cost for some cryptographic primitive.
Let $n$ and $a$ be positive integers ...
- 78.3k
0
votes
1
answer
277
views
Number of possible combinations of X numbers that sum to Y where the order doesn't matters
I am looking for the number of possible outcomes given to a set of numbers X that sum to Y. This is the same issue as here. However, I would like to consider that (i) the numbers can't be repeated and ...
- 320