Not sure if this is possible to calculate in an easy way but given binary string of length n with k bit sets, how many permutations have x consecutive ones and no y consecutive ones where y > x
An example to make sense of what I mean
n = 5, k = 3 we have following permutations
00111 01011 01101 01110 10011 10101 10110 11001 11010 11100
In this example we have for x = 3
00111
01110
11100
x = 2
01011 01101 10011 10110 11001 11010
x = 1
10101
So answer is for x = 3 we have 3 ways, x = 2 we have 6 ways and x = 1 we have 1 way. Can this be expressed within a formula?
Hope I explained it well enough. Cheers