Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.


Suppose I am drawing *k* numbers out of *n* numbers labeled from *1 to n*.
Considering all $\binom{n}{k}$ combinations of numbers drawn, how often does the **maximal difference q** between two consecutive numbers – but also between zero and the lowest number, or the highest number and n – occur.


I already found an algorithm to compute the sequence, but it's too computationally intense for large *n*, so I'm looked for an explicit formula. If it's too complicated to find a formula a distribution would also be fine.


I think someone must already have worked on this problem, but I can't find anything.


The resulting series is:  

(Read: n,m:: q:number of combinations with q as maximal difference)  

2,1:: 1:2  

3,1:: 1:1, 2:2  

3,2:: 1:3  

4,1:: 1:0, 2:2, 3:2  

4,2:: 1:3, 2:3  

4,3:: 1:4  

5,1:: 2:1, 3:2, 4: 2  

...  

10,1:: 8:2, 9:2, 5:2, 6:2, 7:2  

10,2:: 3:3, 4:12, 5:12, 6:9, 7:6, 8:3  

10,3:: 2:4, 3:36, 4:40, 5:24, 6:12, 7:4  

10,4:: 2:45, 3:90, 4:50, 5:20, 6:5  

10,5:: 1:6, 2:120, 3:90, 4:30, 5:6  

10,6:: 1:35, 2:126, 3:42, 4:7  

10,7:: 1:56, 2:56, 3:8  

10,8:: 1:36, 2:9  

10,9:: 1:10 


Where the problem arose:
I'm doing a masters thesis in bioinformatics on a quick clustering algorithm.
So I'm looking for shared *q*-grams (substrings with length q) of pairs of sequences with the length *n* that differ in at most *m* sites.
I want to find the biggest *q* possible so that 99% of all sequences of length *n* with *m* randomly distributed differences share a substring of length *q*.


Illustration of the problem:
(n=10, m=3; X for mismatch, "." for match)



```
X....X...X -> 4
..X..X..X. -> 2
.......XXX -> 7

```

Here's the python script:



```
import itertools
def f(n,m):
    rdict={}
    for d in itertools.combinations(range(n),m):
        t=[-1]+list(d)+[n]
        m=max([x[0]-x[1] for x in zip(t[1:],t[:-1])])-1
        if not rdict.has_key(m):
            rdict[m]=1
        else:
            rdict[m]+=1
    return rdict

```


Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence
Choices of $k$ out of $n$ correspond to ordered $k+1$-tuples of nonnegative numbers which add up to $n-k$ by counting the dots between the Xs.


The number of such $k+1$-tuples so that $a$ particular terms are at least $q$, with no restrictions on the others, is $n-aq \choose k$ [edit: when $n-aq \ge 0$, and $0$ otherwise], since subtracting $q$ from each of the terms we know are at least $q$ gives an unrestricted $k+1$-tuple adding to $n-aq$. So, the technique of inclusion-exclusion lets us count $f(n,k,q)$, the number of $k+1$-tuples with no term which is at least $q$:


$$ f(n,k,q) = \sum\_{a=0}^{k+1} (-1)^a {k+1\choose a}{n-aq\choose k}.$$


To count sequences where the maximum is exactly $q$, take $f(n,k,q+1)-f(n,k,q)$.




---


Edit: The above formula is incorrect because it includes the terms where $n-aq$ is negative. For these terms, replace $n-aq \choose k$ with $0$, or change the upper limit of the sum:


$$ f(n,k,q) = \sum\_{a=0}^{\lfloor n/q \rfloor} (-1)^a {k+1\choose a}{n-aq\choose k}.$$


