Here and in what follows Bin(m) denotes a binomial random variable with parameters m and p = 1/2.
The above proof is typical for our arguments : we will condition, use (4), compute an expectation involving a function of a binomial random variable, and finally use (5) to reduce the length of the tableaux by one.
The relevant expectation involving a binomial random variable is isolated in the following lemma (observe that b - 1 and c - 0 gives the property used in the proof of Proposition 3).
The relevant property of a binomial random variable is
We will need the following property of a binomial random variable.
2008, "Ordering comparison of negative binomial random variables with their mixtures", Statistics and Probability Letters.
2003, "Stochastic comparisons of Poisson and binomial random variables with their mixtures", Statistics and Probability Letters.