# Minimize number of lattice paths below a given path

Every north-east lattice path (NE-path) $$v$$ from $$(0,0)$$ to $$(k, a)$$ can be identified with a sequence $$0 \le \lambda_1 \le \lambda_2 \le . . . \le \lambda_k\le a$$, that represent the hight of each step. There is a well known formula for the number of NE-paths from $$(0,0)$$ to $$(k, a)$$ below the $$v$$, namely $$\ell(v)= \det_{1 \le i,j \le k} \left( \binom{\lambda_i+1}{j-i+1} \right).$$ Now my question is, for a fixed endpoint $$(k,a)$$, and a fixed number $$n>a$$, is it known which shape $$0 \le \lambda_1 \le \lambda_2 \le . . . \le \lambda_k$$ with $$\lambda_1+ \dots + \lambda_k+a=n$$ that gives the smallest value for $$\ell(v)$$? I. e. which shape should I give my path, so that there is as few paths below it as possible?

• I do not understand the question: isn't there just one path for $\lambda_1=\dots=\lambda_{k-1}=0$ and $\lambda_k=n$? Or did you mean to set $\lambda_k = k$? – Martin Rubey May 11 at 14:10
• Oops, good point! I meant also to fix the endpoint. I will edit the question. @MartinRubey – Mari May 11 at 15:22
• @Mari possibly you mean $\lambda_1+\ldots+\lambda_k=n$, without $a$? This sounds more natural (we fix the area of the Young diagram) for me. – Fedor Petrov May 11 at 22:34