binomial determinant
Edwin Clark
eclark at math.usf.edu
Sun Sep 14 05:47:31 CEST 2003
Benoit,
In looking at det([binomial(i*z_j,j)]) it seemed to me that it would help
to know det([binomial(a_i*z_j,j)] ) which appears to be related to the
vandermonde determinant. For example I get the following generalization
of your determinant using Maple:
det([binomial(a_i*z_j,j)]
= 1/(C(n))* prod(a_i,i=1..n)*prod(a_i-a_j, 1<=j<i <=n)*prod(z_j^j,j=1..n)
where C(n) = A000178(n) = the Vandermonde determinant of the numbers
1,2,..(n+1) (among other things).
I haven't proved it, but Maple verifies it symbolically up to n = 5 and
putting in specific numbers it checks out up to n = 8.
--Edwin
On Sat, 13 Sep 2003, benoit wrote:
>
> 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
> >
> >
>
------------------------------------------------------------
W. Edwin Clark, Math Dept, University of South Florida,
http://www.math.usf.edu/~eclark/
------------------------------------------------------------
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