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
```