Finding total number of self avoiding paths for $n\times n$ grid

Question

we call a connected part of $n\times n$ grid "N-mino" if it has these 2 conditions

1. it should contain $(n,n)$
2. if it contains $(i,j)$ then it should contains at least one of $(i+1,j)$ or $(i,j+1)$.

Find number of N-mino for a $n\times n$ grid.

For example: if $n=1$, there's just 1 such path , if $n=2$ there are 7 paths. thnq in advanced ^_^.

Answer 1

What you call "N-minos" are normally called directed polyominoes. There is quite a large literature on them. Perhaps this is a useful general starting point.

However, I'm not aware of them being counted by bounding box size before (usually by size or perimeter). Using Mathematica, it appears the first few terms of the series are $1, 7, 79, 2007, 131311$.