Help Analytic combinatorics

[lfccruz]

Hello,

I'm not a math student, but I need help for the following situation (example):

Have a vector of size n (data structure), filled with values 1:
Example: n = 12
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

I have sequences of zeros of length m.
Example: m = 1
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |

Example: m = 2
| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |

Example: m = 3
| 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |

How many combinations are there for a vector of n positions and sequences of zero values up to size m?
Example: For n = 12 and m = 3.

no sequence of zeros
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

combinations of sequences x 1 zeros
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |

combinations of two zero sequences x
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 |

combinations of three zeros sequences x
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |

combinations of sequences of zeros of different sizes
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |

I need the mathematical formula (formal) and a way to get one by one the possible combinations!!

I do not know if I created the topic in the correct place.
Sorry for my terrible english because I speak Portuguese (Brazil).

Thank you!
Luiz Fernando

[Guest]

This is just the number of length [tex]n[/tex] sequences of 0-1 with no [tex]m + 1[/tex] stretch of 0s. One of those is up to [tex]m[/tex] times 0, then 1, followed by one of the sequences of interest.

[aron123]

Thank for sharing....