Combinatorics problem about determining number of paths

[Sprooth]

Hi,

I'm trying to figure out a variation of the problem where you determine the number of paths on a discrete grid. For example, for the paths from (0,0) to (a,b), you can consider it to be a rearrangement of a word with a x's and b y's, so the number of paths is (a+b)! / a!b!.

The variation I would like to figure out would allow a specified number of reverse movements in addition to the movements in the proper direction. I am representing a movement to the right by "x", a movement to the left by "w", a movement upward by "y", and a movement downward by "z". If I am finding a path from (0,0) to (a,b) with c reverse horizontal movements and d reverse vertical movements, then the number of paths I have so far would be:

(a + b + 2c + 2d)! / (a+c)!(b+d)!c!d!

since the paths could be considered as a rearrangement of a word made of a+c x's, b+d y's, c w's and d z's. (a+c x's since there must be an additional movement to the right for every allowed movement to the left.

I believe this is correct so far, but now I want to prevent paths where for example it goes right, then left, then right with a net result of just going right. To do this I want to find rearrangements of the word where there are no adjacent x's and w's and no adjacent y's and z's. I began by simplifying the problem to one where only reverse horizontal movements were allowed and I would then attempt to generalize to the case with reverse movements in both dimensions. However, I got stuck even in the simpler case.

Would someone be able to help me through the simpler case (where there would be x's for going right, y's for going up, and w's for going left)? Then I will try to generalize to the other case.

Thanks for your help.

 

[mighty2000]

Hello,

Thank you for your inquiry. This is an interesting variation of the problem and it seems like you have made some good progress in your approach. To address the issue of preventing paths with adjacent x's and w's, you could consider using a technique called "inclusion-exclusion principle". This principle can be used to count the number of objects that satisfy certain conditions, while excluding those that do not.

In this case, you could consider the paths that have adjacent x's and w's as one group, and the paths that do not have adjacent x's and w's as another group. Using the principle, you can then subtract the number of paths with adjacent x's and w's from the total number of paths, to get the number of paths without adjacent x's and w's.

For the simpler case where only reverse horizontal movements are allowed, you could approach it by considering the paths that start with a reverse movement (w) and those that do not. You could then use the inclusion-exclusion principle to count the number of paths with no adjacent x's and w's. Generalizing this approach to the case with reverse movements in both dimensions may require some additional considerations, but it could be a good starting point.

I hope this helps. Let me know if you have any further questions or if you need any clarification. Good luck with your research!