All Questions
10
questions
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
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 ...
- 478
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 ...
0
votes
0
answers
51
views
Summation of a prime and a prime power
Is there an even number $n \in \mathbb{N}$ and two different primes $p,q<n$ which are not divisors of $n$, as well as $a,b \in \mathbb{N}$ with $a,b>1$, such that
$$
n=q+p^{a}=p+q^{b}
$$
? I ...
- 21
3
votes
1
answer
85
views
Showing $\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}$
I want to show
\begin{align}
\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}
\end{align}
I know one proof via self-conjugation of partition functions with ...
- 6,490
0
votes
1
answer
307
views
Get combination of numbers that when added same as the given number
For a given number $n >0$ is there a way to get combination that add up to this number??
for example :
if $n=6$ then numbers that add up are $5+1,4+2,3+2+1$ so the combination is 3
if $n=4$ then ...
- 103
2
votes
0
answers
63
views
Closed-form solution of sum over compositions?
I am interested in calculating a closed-form solution of the following sum over compositions $$ \sum_{\substack{n_1 + \dots + n_M = N \\ n_i \geq 1}} \dfrac{n_1^2 + \dots + n_M^2}{n_1(N-n_1)! \dots ...
1
vote
0
answers
121
views
Expressing a sum over the sizes of the parts of every partition of n
Let $(a_1^{r_1},\ldots,a_{p}^{r_{p}})\vdash n$ be the multiplicity representation of an integer partition of n. Each $a_{i}$ is a part of the partition and $r_{i}$ is its corresponding size. We ...
- 1,401
1
vote
0
answers
76
views
Counting number of integer solutions to $a_1 + a_2 + a_3 + \ldots = n$ where all $a$'s must be in certain range
For a given $(n,m,k)$..
Using values in the range $(0..k)$, how many different $m$-combos exist which sum to n?
ex. for $(n,m,k)$ = $(3,3,2)$, there are 7 possible combinations. For $(5,4,2)$ ...
- 11
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 ...
- 293