Feel free to have a look at a preprint on my public github repo if you like. It is enumerative combinatorics and stats and lattice geometry.

The biggies in prior reading were Mann, Whitney, Fisher, Stanley, Knuth, De Bucchianico, Cummings and u/efrique and u/edderiofer.

It seemed open to me how many paths there are across a lattice. For Manhattan paths we can look to Pascal’s triangle and the binomial. But what about diagonals? At first glance it’s gnarly and combinatorially explosive.

Here is my answer, which builds a triangle quite like Pascal’s but with a different recursion. It’s on an open repo under Creative Commons and it is submitted. Main challenge for me has been trying to write a clear proof - I am a doctor to trade awaiting an applied stats/policy doctoral viva.

I don’t always fanfare submissions but I think the new triangle, which is called McMeekin Hill instead of Pascal’s Triangle (after my fantastic PhD advisors) may have rich properties like Pascal’s that I cannot exploit. I am not jealous about exploring them and some hardcore pure maths person might benefit. I know Erdos did some things with Goldbach and Binomial for example but if I continue to follow the rabbit I will lose balance.

The distribution is called the “Occlusion” distribution not “Binomial” and the related function is called “See” not “Choose”. Clearly if anything smells bad I’d rather know.

Repo is here:

https://github.com/keithreid-sfw/McMeekinHillDiagonalsLattice