All Questions
7
questions
0
votes
0
answers
19
views
Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
- 91
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
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 ...
6
votes
0
answers
141
views
Faa di Bruno's formula and alternating functions
Suppose you have a function $f(x)$ such that ${\rm sgn}\Big(\frac{d^k}{dx^k}\big(f(x)\Big) = (-1)^k$ and a function $g(x)$ such that ${\rm sgn} \Big(\frac{d^k}{dx^k}g(x)\Big) = (-1)^{(k+1)}$, $\forall ...
- 251
2
votes
1
answer
4k
views
Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct
This is from a previous question paper for an entrance exam I am preparing for.
https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
- 53
4
votes
1
answer
3k
views
How many ways to write a number $n$ as the product of natural numbers $\geq 2$?
I am looking for a closed form (or efficient algorithm) for $f(n)$, the number of ways in which $n$ can be written as a product of natural numbers $\geq 2$. Note that $f(n)=\sum_{i=1}^{\Omega(n)}{g(n,...
- 535