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?

  • $\begingroup$ 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 $\lambda_k = k$? $\endgroup$ – Martin Rubey May 11 at 14:10
  • $\begingroup$ Oops, good point! I meant also to fix the endpoint. I will edit the question. @MartinRubey $\endgroup$ – Mari May 11 at 15:22
  • $\begingroup$ @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. $\endgroup$ – Fedor Petrov May 11 at 22:34

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