A fun combinatoric puzzle that's popped up in my work that I think would be cute to have a Mathematica solution to, if anyone wants to give it a go. It's basically a ladder climbing/descending problem so probably has a nice
Graph solution. It is worth noting that my ladder can descend into the basement (i.e. my integer values can go below $0$)
Starting at $0$, over $k$ steps of $\pm1$, what are the paths that will land on the integer $n$, assuming of course that $k \ge n$.
I don't mind if this question gets closed for lack of effort on my part (I'm currently working out an analytic solution) and would actually be very happy if this were closed as a duplicate/if someone could point me to the proper name for this problem. But I thought Mathematica.SE might enjoy a quick, easy problem to break the "solve my integro-differential equation for me" drudgery.
m = (k - n) / 2is an integer number, and then the total number of such distinct paths should just be
C(k, m) = k!/(m!(k-m)!). $\endgroup$
mare both even/odd. I'm working on getting the paths themselves, not just the number of them, unfortunately. $\endgroup$