Matrix of values generated randomly with Univariate Distribution I am trying to understand an observation made in a research paper. Before i write down the observation i want to put the context to avoid any confusion. The observation is about a matrix of values and in that matrix the values, which are somehow related to each other, need to be grouped together and form a new sub matrix. Here is the observation.  
Given the input matrix $ A= \{ a_{ij} \} $ where $i$ stands for rows and $j$ stands for columns, consider counting the number of rows with the property that the value in column $p$ is smaller than the value in column $q$ i.e. $\#\{k: a_{kp} < a_{kq}\}$ . If the values in the dataset are generated randomly with univariate distribution, half of the rows on average are expected to have this property and half are not. Addition of another column $r$ to the series, such that values in this column $r$ are larger than the values in column q i.e $\#\{k: a_{kp} < a_{kq} < a_{kr}\}$ should result in another reduction of number of rows by half . Thus, addition of the column to the series reduces the number of concordant rows by half
Now my first question arises here, Do i need to iterate over $i$ and $j$ to select $p$ and $q$. What strategy is being used here to select the values of $p$ and $q$. Secondly can you give me a square matrix of $8$x$8$ that has univariate distribution and i can apply both of the said property. Lets continue the observation.  
On the other hand is the distribution of the data is not uniform and there exists a monotonic relationships between rows in some subsets of conditions any addition of the pattern specific column wont eliminate the rows belonging to this pattern. 
Now what kind of monotonic relationship will yield this type of property. Is there matrix that helps to understand this. Wasnt the property $\#\{k: a_{kp} < a_{kq} < a_{kr}\}$ had some sort of relationship too.
I know thats a bit long post but i really need to understand this observation for the thesis defense. This paragraph is taken from the paper named as EBIC: An evolutionary based parellel biclustering algorithm for pattern discovery
 A: What that paragraph is saying is "suppose we have the matrix, and pick any two indices $p$ and $q$." So maybe it's a $12 \times 12$ matrix, and I look at $p = 4$ and $q = 7$. 
Now walk through the rows, one at a time. Is $a_{13} < a_{17}$? The answer's either "yes" or "no". Is $a_{23} < a_{27}$? Again, yes or no. When you're done counting up, you'll find some number of rows produced a "yes" answer --- maybe five of them. In that case, $\#\{k : a_{kp} < a_{kq} \}$ is five. But on average, given the uniform distribution of the $a_{ij}$, you'd expect it to be six. Maybe for some other $pq$ pair, you'd get seven. But the average number of rows with this property, for a fixed $p,q$ pair, should be six. 
The statement following this one, "Addition of another column $$ to the series, such that values in this columns are larger than the values in column $q$ ..." doesn't seem entirely obvious to me; the numbers in column $r$ don't have to be larger than those in some random position, but rather must be larger than those in column $q$, which is already supposed to be larger than column $p$, so we could have to compute this as a conditional probability, and I have my doubts about the glibness of the assertion that we lose a factor of two...but I could easily be wrong. I haven't even had a cup of tea yet this morning. 
The "On the other hand..." sentence doesn't really make sense to me, because it talks about "conditions", which might index rows or columns, which I can't discern without reading more of the paper. 

I wanted to check my suspicions about the conditional probability, so I wrote some matlab code: 
function trand()
trials = 2000;
u = zeros(trials, 1);
v = zeros(trials, 1);
for s = 1:trials
  N = 24;
  a = rand(N);
  p = 1; q = 2; r = 3;
  v(s) = sum( a(:, p) < a(:, q));
  u(s) = sum( (a(:, p) < a(:, q)) & (a(:,q) < a(:,r)));
end
mean(v)
mean(u)

When I ran this code, I got that the average number of rows of my $12 \times 12$ matrix for which the $p$th entry is less than the $q$th was $12.0195$, as expected and claimed in the paper. (That was mean(v) in the program.) When I tried adding in the $r$th column, I got mean(u) turning out to be $4.0280$, which is a long way from the $6$ that the authors seem to be predicting. 
Frankly, either I'm misreading what they're claiming, or they're just plain wrong. 

Actually, after a few more minutes of thinking, I'm pretty sure they're wrong. For in picking the indices $p, q, r$, I could have made any one of six choices of order; each of those six choices of order should result in the same expected number of rows. But that means that the expected number of "good" rows for any one choice should by $1/6$th of the total number of rows, i.e., $4$, which my numerical experiments confirm, and not $6$. 
A: I am the author of this paper. You are absolutely right, Asian, there is a glitch in the paper that makes this particular claim false. The rest of the paper is not impacted at all, as the fitness function could be defined in any way. 
The correct version of this paragraph should be:
Addition of another column  to the series, such that values in this column  are larger than the values in column q i.e $\#\{:_{}<_{}<_{}\}$ should result in another reduction of number of rows by at least half. Thus, the addition of the column to the series reduces the number of concordant rows by at least half. 
The actual value that describes this property (i.e. the number of rows that follow a particular monotonous order of length k) is the number of rows of the dataset divided by the factorial of k.
