Asymptotics for the Distributions of Subtableaux in Young and Up-Down Tableaux

  • David J. Grabiner


Let $\mu$ be a partition of $k$, and $T$ a standard Young tableau of shape $\mu$. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of $N$ cells contains $T$ as a subtableau is asymptotic to $f^\mu/k!$ as $N$ goes to infinity, where $f^\mu$ is the number of all tableaux of shape $\mu$. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young tableaux with at most $n$ rows is proportional to $\prod_{1\le i < j\le n}\bigl((\mu_i-i)-(\mu_j-j)\bigr)$; as $n$ goes to infinity, the probabilities approach $f^\mu/k!$ as expected. We have a similar formula for up-down tableaux; the probability approaches $f^\mu/k!$ if $\mu$ has $k$ cells and thus the up-down tableau is actually a standard tableau, and approaches 0 if $\mu$ has fewer than $k$ cells.