explicit forms, that can be obtained from a general formula 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 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 where ˆ τ is the partition that consists of r identical parts ( p − r + 1) /α − 1 . Finally, note that Therefore, ˜ γ ( α $^{)}$=1 and the statement of the lemma follows from (6) and (38). □ Proposition 1 and Corollary 1 directly imply (18) and the following formula for Proof of Proposition 2 26 ∏ s ∈ ˆ τ ( r +( a $^{′}$( s )+1) α − l $^{′}$( s ) − 1) = α | ˜ τ |− t ∏ s ∈ ˆ τ (( r − l $^{′}$( s ) − 1) /α + a $^{′}$( s )+1) = α | ˜ τ |− t r ∏ j =1 Γ (( r − j ) /α +( p − r +1) /α ) Γ (( r − j ) /α +1) = α | ˜ τ |− t r ∏ j =1 Γ (( p − j +1) /α ) Γ (( r j ) /α +1) . ˜ γ ( α ) = α | ˜ τ | α t r ∏ j =1 Γ (( p +1 − j ) /α ) Γ (1+( r j ) /α ) 1 ∏ s ∈ ˆ τ ( r +( a $^{′}$( s )+1) α − l $^{′}$( s ) − 1) , ˜ γ ( α ) = α | ˜ τ | α t r ∏ j =1 Γ (( p +1 − j ) /α ) Γ (1+( r j ) /α ) ∏ s ∈ τ ( p + αa $^{′}$( s ) − l $^{′}$( s )) ∏ s ∈ ˜ τ ( r +( a $^{′}$( s )+1) α l $^{′}$( s ) − 1) . C ( α ) κ ( I$_{m}$ ) = α | κ | | κ | ! w ( κ, α ) ∏ s ∈ κ ( m + αa $^{′}$( s ) − l $^{′}$( s )) . (39)