binomial determinant
benoit
abcloitre at wanadoo.fr
Sat Sep 13 17:14:00 CEST 2003
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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