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

Let μ be a partition of k, and T a standard Young tableau of shape μ. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of N cells contains T as a subtableau is asymptotic to fμ/k! as N goes to infinity, where fμ is the number of all tableaux of shape μ. 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 ∏ 1≤i<j≤n ( (μi − i) − (μj − j) ) ; as n goes to infinity, the probabilities approach fμ/k! as expected. We have a similar formula for up-down tableaux; the probability approaches fμ/k! if μ has k cells and thus the up-down tableau is actually a standard tableau, and approaches 0 if μ has fewer than k cells.