content_image/299437.jpg content_image/299436.jpg content_image/299435.jpg content_image/299434.jpg
explicit forms, that can be obtained from a general formula 

<img src='content_image/299437.jpg'>

A variant of this formula, that uses the generalized Pochhammer symbol, can be found, for example, in Dumitriu et al (2007, Table 5). Then, after cancellations, we get 

<img src='content_image/299436.jpg'>

Now consider the last ratio of the products in the above expression. For the product term in the numerator that corresponds to square s in the position ( i, j ) in the diagram of τ, there exists exactly the same term in the denominator, which corresponds to square s in the position ( i, j + ( p − r + 1) /α − 1) in the diagram of ˜ τ . Therefore, we can write 

<img src='content_image/299435.jpg'>

where ˆ τ is the partition that consists of r identical parts ( p − r + 1) /α − 1 . Finally, note that 

<img src='content_image/299434.jpg'>

Therefore, ˜ γ ( α $^{)}$=1 and the statement of the lemma follows from (6) and (38). □ 

## Proof of Proposition 2 

Proposition 1 and Corollary 1 directly imply (18) and the following formula for 

