co.combinatorics – Combinatorics: Distribution of red, blue and green bullets in separate bins

Suppose there are three types of balls: red R balls, blue B balls and green G balls. How many ways are there to distribute the bales in N different bins so that no bin is empty and that a bin can hold no more than one balloon of the same color?
Right here,

1) R, B, G <= N (no color should have more than N balls, because we have the condition that at most one ball of the same color is in a tray and that all balls must be placed )

and

2) R + B + G> = N (this is due to the fact that no basket is empty).

Combinatorics – Arrange the letters of the alphabet into four strings to generate specific permutations

The English alphabet contains 26 letters. How many chains of four different letters are there:

a. in total?

b. in which all the letters are distinct?

c. which do not contain the letter a?

re. which contain the letter at least once?

e. which contain the letter at least once and the letter b at least once?

F. which contain the letters a and b in consecutive positions with a previous b, and all the letters are distinct? [Examples include abcd, dabf, and ghab, but not acbd, bagd, or abbb.]

combinatorics – Combination Help – Mathematics Stack Exchange

Suppose that there is an arbitrarily large number of copies of ten separate math books (which we will call titles) for cmsc250. Each student must choose a title as a textbook. Suppose the five hundred students taking the course are indistinguishable

How many different ways can students choose a title, if Epp should be chosen by at least twenty students and if Rosen should be chosen by at least thirty students?

I had $ {459} choose {10} $, I subtracted (20 + 30) from the total number of students (500) and then, using the formula $ C (n + r-1, r) $ I have connected the corresponding numbers. Is it correct? If not, can anyone help me find the right path?
Thank you

combinatorics – How to generate all combinations

There is $ N $ optimization variables, $ v_1, v_2, cdots, $ v_N.
and $ v_n in {0,1,2,3, cdots, K} $.

Let $ N = $ 10 and K $ = $ 5.

How can I generate all possible combinations?

For example, the first combination is $[0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0hspace{1mm} 0]$

The last combination is $[5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5hspace{1mm} 5]$

combinatorics – Will Shor's algorithm break a permuted list in post-quantum?

If I create a list of 100 random numbers – create a hash from the list – swap the list – then give you the swapped list and hash from the original list … it would take you a while to find the control of the origin of the numbers in the list … since this list would have approximately 9.3326215443e + 157 possible permutations (unauthorized repetitions).

Will the Shor algorithm make it easier to search for the original order of such a permuted list once quantum computers become available?

combinatorics – properties of identically auto-dual matroids

I'm dealing with a matroid M identical to the dual on the vertices E =[2N]In other words, if B is a base of M, E-B is also a base of M itself. I need simple combinatorial properties of these elements, such as the structure of the flats network, the properties of the basic circuits of the bases, and so on. Nothing important, but before you spend a lot of time there, does anyone know if I find this systematically done somewhere?

combinatorics – Complex query to identify a subset

Fix an ambient finished set $ X $without loss of generality $ X = {1, ldots, | X | } $. You want to identify a subset $ B subseteq X $ hidden to you. You can get information about the subset as follows: you can choose any subset. $ C subseteq X $and you receive the number $ | B cap C | $.

In terms of $ | X | $ (and eventually $ | B | $), what is the minimum number of queries needed to determine $ B $? How about a particular case, for example when $ | B | leq log_2 (| X |) $?


Some observations:

  • L & # 39; s case $ | B | = $ 1 allows for binary search, and so can be done in $ lceil log_2 (| X |) rceil $. Already for $ | B | = $ 2, a way with less $ 2 log_2 (| X |) + C $ queries are not clear to me (although I can see how to do $ frac32 log_2 (| X |) + C $ on average).
  • trivially $ | X | $ queries are always enough: just start with $ C = X $ and delete an item for each query.
  • This question was inspired by a recent Google Code Jam problem: your query consists of one bit for each element of $ X $and you get $ | X | – | B | $ bits back. However, in the Google Code Jam problem, the returned bits are always sorted correctly, but with the missing values. This allows you to match responses from multiple queries. then $ lfloor log_2 (| B |) + 1 rfloor $ the requests are sufficient.

combinatorics – algorithms to generate a random rectangle without defects?

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I want to generate a random rectangular partition of a given $ m * n $ rectangle under the constraint that it must act from a faultless partition. Partition without defect means that a dissection of a rectangle into smaller rectangles so that the original rectangle is not divided into two sub-rectangles

Basically, there are not two adjacent rectangles that share a common side and at most two corners meet at one point. These partitions appear under other names including Rectangulars, Tatami Tiles and Uncut Ground Plans.

What are the algorithms that generate random rectangular partitions without defects?

Note: This was first posted to Mathematics SE without response.

combinatorics – software metrics for data growth

I write an article for software that uses combinatorics to generate large sets of results. I would like to describe that if I put in n elements, I will have in return 2 ^ n elements.

Is there a software metric describing the "data growth" of an algorithm or should I just write "data exponentially increase"?

btw: The size of the 2 ^ n elements varies, so I can not determine their size (storage)

combinatorics – Search for the optimal bijective map between two sets of natural numbers

I have two sets $ A $ and $ B $, each containing $ N $ large vectors.
I have a $ N times N $ matrix, $ M $, or $ M_ {i, j} = text {Dist} || A_i, B_j || $.

I'm trying to find a bijective map $ A rightarrow B $ which minimizes the average distance between paired elements.

The surjective case is obviously trivial; just the map $ A_i $ at $ B_j $ or $ j = text {argmax} (M_ {i, j}) $. However, I want a 1-to-1 mapping.

$ N $ is not very big (~ 5), but I have many pairs of this type and I need to find an algorithm to automate the process.

Any help appreciated!