binomial determinant

[benoit]

Thanks for the confirmation. I wonder if the following Gessel and 
Viennot paper on binomial determinants could help.

I.Gessel and X. Viennot, Binomial determinants, paths and hook length 
formulae , Adv. in Math., 58 (1985), 300–321

I didn't read it but I know that's a nice interpretation for binomial 
determinants.

Benoit.

>>
>> let z be any function : N-->N, let M_n be the nxn matrix
>> M_(i,j)=binomial(i*z(j) , j),  then I observed :
>>
>> det M_n = prod(k=1,n, z(k)^k)
>>
>> I'm looking for  a proof or anything related... Thanks.
>>
>> Benoit Cloitre
>>
>
> Maple gives a proof for n from 1 to 5:  I replace your functional 
> values
> z(j) by inteterminates z_i (or z[i] in Maple notation). Also I convert 
> the
> binomial expressions to factorial expressions. Then it holds without
> restriction on the variables:
>
>> with(LinearAlgebra):
>> for n from 1 to 5 do
>> M:=Matrix(n,n,(i,j)->convert(binomial(i*z[j],j),factorial)):
>> simplify(Determinant(M));
>> print(n,%);
>> end do:
>
>                                1, z[1]
>
>
>                                         2
>                             2, z[1] z[2]
>
>
>                                      2     3
>                          3, z[1] z[2]  z[3]
>
>
>                              4          2     3
>                       4, z[4]  z[1] z[2]  z[3]
>
>
>                           4     3     5     2
>                    5, z[4]  z[3]  z[5]  z[2]  z[1]
>
>
> With some effort there should be a proof of this. :-)
>
> --Edwin
>
>
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