Intereting Posts

A practical way to check if a matrix is positive-definite
Prove that the empirical measure is a measurable fucntion
Can we found mathematics without evaluation or membership?
Decomposition Theorem for Posets
Two possible definitions of “vector-valued distribution”
Show $\int_{0}^{\frac{\pi}{2}}\frac{x^{2}}{x^{2}+\ln^{2}(2\cos(x))}dx=\frac{\pi}{8}\left(1-\gamma+\ln(2\pi)\right)$
Another question on almost sure and convergence in probability
Map of Mathematical Logic
A type of local minimum (2)
Is $|f(a) – f(b)| \leqslant |g(a) – g(b)| + |h(a) – h(b)|$? when $f = \max\{{g, h}\}$
What does it mean for a function to be a solution of a differential equation?
Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3$
Rotating x,y points 45 degrees
Integral involving $\coth (x)$: Maple and Mathematica disagree
Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 – x} + \sqrt{x}}dx$

Suppose that, for a real application, I have ended up with a sorted list A = {$a_1, a_2, …, a_{|A|}$} of elements of a certain kind (say, Type-A), and another sorted list B = {$b_1, b_2, …, b_{|B|}$} of elements of a different kind (Type-B), such that Type-A elements are only comparable with Type-A elements, and likewise for Type-B.

At this point I seek to count the following: in how many ways can I merge both lists together, in such a way that the relative ordering of Type-A and Type-B elements, respectively, is preserved? (i.e. that if $P_M(x)$ represents the position of an element of A or B in the merged list, then $P_M(a_i)<P_M(a_j)$ and $P_M(b_i)<P_M(b_j)$ for all $i<j$)

I’ve tried to figure this out constructively by starting with an empty merged list and inserting elements of A or B one at a time, counting in how many ways each insertion can be done, but since this depends on the placement of previous elements of the same type, I’ve had little luck so far. I also tried explicitly counting all possibilities for different (small) lengths of A and B, but I’ve been unable to extract any potential general principle in this way.

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- numbers from $1$ to $2046$
- Enumerate certain special configurations - combinatorics.
- Strehl identity for the sum of cubes of binomial coefficients
- Weird $3^n$ in an identity to be combinatorially proved

- The number of (non-equal) forests on the vertex set V = {1, 2, …,n} that contains exactly 2 connected components is given by
- In how many different ways can boys and girls sit a desks such that at each desk only one girl and one boy sits?
- Finding the probability that red ball is among the $10$ balls
- Placing checkers on an m x n board
- Combinatorics - Integer sided triangles with integer median
- how many ways can the letters in ARRANGEMENT can be arranged
- How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$
- SAT Probability and Counting
- There exists a vector $c\in C$ with $c\cdot b=1$
- Traveling salesman problem: a worst case scenario

The merged list has length $|A|+|B|$. Once you know which $|A|$ positions in it are occupied by the elements of list A, you also know exactly how the whole thing has to be ordered, since the internal orders of the elements of A and the elements of B are already known. Thus, there are $$\binom{|A|+|B|}{|A|}=\binom{|A|+|B|}{|B|}$$ possible merged lists, one for each choice of $|A|$ positions for the elements of list A.

$\frac{(m+n)!}{(m)!(n)!}$;

where m,n is the length of the arrays.

$(m+n)!$ is the total number of ways of arranging without any restriction.

The condition given that you can not change the order of each array so you should not change the order of each array , so we need to divide all the permutations of each array!

- Krull dimension and transcendence degree
- Does a convergent power series on a closed disk always converge uniformly?
- Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units
- Proof of triangle inequality
- Is there a simpler way to find an inverse of a congruence?
- Property of $\dfrac{\sum a_i}{\sum b_i}$ when $\dfrac{a_i}{b_i}$ is increasing
- Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $
- Prove that if $f(n) \in O(g(n))$ then $g(n) \in \Omega(f(n))$
- Preserving structures
- Are there open problems in Linear Algebra?
- Multiplying three factorials with three binomials in polynomial identity
- Bounded linear operator maps norm-bounded, closed sets to closed sets. Implies closed range?
- Question about a proof in Rudin's book – annihilators
- Parametrization of $x^2+ay^2=z^k$, where $\gcd(x,y,z)=1$
- Differential Operator Issue