All Questions
9
questions
1
vote
2
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184
views
Sum of the product of Permutations
I found a similar problem on one of my problem sets and am struggling with how to approach the problem.
The problem is as follows:
$$
\text{Evaluate} \sum _{k=1}^{n}\:P(n, k-1)P(n-1,n-k).
$$
The ...
3
votes
2
answers
178
views
What is $\underset{\pi \in S_n}{\max}\underset{1\leq i\leq n}{\sum}|i-\pi(i)|$, where $S_n$ is the set of permutations of $(1,2,\ldots,n)$?
What is the value of $$\underset{\pi \in S_n}{\max} \biggl( \underset{1\leq i\leq n}{\sum} \big|i-\pi(i)\big|\biggr)\,,$$ where $S_n$ is the set of permutations of $(1,2,\ldots,n)$?
I can see that if ...
- 704
1
vote
0
answers
80
views
Summing discrete coordinate lengths, a generalized n-dimensional case
A previous question pertains to a formula for the total number of points in a 3d discrete coordinate system that each total number of coordinate digit lengths $l$ can describe.
Is it possible to ...
- 33
2
votes
1
answer
100
views
Summing discrete 3d coordinate lengths
Consider a three dimensional discrete coordinate system $(x,y,z)$, where $x,y,z\in$ natural numbers.
The number of digits describing an integer coordinate for each dimension is $l_c=\lfloor log(c) \...
- 33
-1
votes
1
answer
423
views
How to calculate the sum of the total set of numbers from a set of digits (without repetition)?
The question seems fairly complicated but let me break it down:
For example, you are to calculate the sum of the possible 4 digit numbers of the digits 1,2,3,4 without repetition. So all the numbers ...
- 111
4
votes
1
answer
836
views
How to expand a summation of "n choose k" into a polynomial?
I wish to expand
$$\sum\limits_{k = 0}^{i - 1} {n \choose k}$$
where $i \leq n$ into a polynomial. However I am uncertain about the coefficients of each term. I have consulted https://en.wikipedia....
- 1,156
0
votes
1
answer
746
views
Proof by induction that $\sum$ $n\choose k $ $= 2^n$ [duplicate]
Trying to prove by induction that
$\sum^{i=n}_{i = 0}$$n\choose i$$=2^n$
So obviously, I prove for some arbitary number, i.e $n = 2$
I then go on to show what the n = k terms look like for the ...
- 91
2
votes
3
answers
288
views
Permutation of integers
Let $n$ be a positive integer and let $(a_1,...,a_n)$ be a permutation of $\{1,2,...,n\}$. Define $$A_k = \{ a_i | a_i < a_k, i >k\} \\ B_k = \{a_i | a_i > a_k, i < k\}$$
for $1 \leq k \...
user72151
1
vote
2
answers
140
views
Prove summation related to cycles: For $n \geq r $ there is:$ \sum_{k=1}^{n} {b_r(n,k)x^k=(r-1)!\frac{x^\overline{n}}{(x+1)^\overline{r-1}}} $
Let $b_r(n,k)$ be the number of n-permutations with $k$ cycles, in which numbers $1,2,\dots,r$ are in one cycle.
Prove that for $n \geq r $ there is:
$$
\sum_{k=1}^{n} {b_r(n,k)x^k=(r-1)!\frac{x^\...
- 1,013