Number of monomials in discriminant of a polynomial of degree n.

[Dean Hickerson]

Neil Sloane wrote:

> Dear seqfans, at present i have this:
> 
> %I A039744
> %S A039744 1,2,5,18,73,338,1656,8512,45207,246448
> %N A039744 Number of monomials in discriminant of a polynomial of degree n.
> %C A039744  Each monomial in the discriminant of a polynomial of degree n
> is an integer constant times the product of 2(n-1) of the coefficients,
> the sum of whose indices (weight) is n(n-1); their number = number of
> ways n(n-1) can be partitioned into the sum of 2(n-1) integers in the
> range 0..n.
...
> A correspondent writes that
> I came upon this sequence in my own work and after repeated checks it seems
> to me that this sequence in fact starts as "1,2,5,16,59,...". Am I wrong?
>
> Could ssomeone check this?

Victor Miller wrote:

> According to Magma, the sequence starts
...
> [ 1, 2, 5, 16, 59, 246, 1103 ]

Tony Noe notes that this is A007878.

I've confirmed the terms up to a(8)=5247.

The current A039744 actually gives the number of ways n(n-1) can be
partitioned into the sum of 2(n-1) integers in the range 0..n.  (I've
checked that for the 10 terms that are given.)  But the claim that this
equals the number of monomials in the discriminant is false.  The first
counterexample is n=4:  There are 18 such partitions, but the discriminant
has no terms corresponding to the partitions 3+2+2+2+2+1 and 2+2+2+2+2+2,
so the number of monomials in the discriminant is only 16.

According to Wikipedia, the discriminant of  a_0 + a_1 x + ... + a_n x^n
is 1/a_n times the determinant of a particular matrix; for n=4 that matrix is

    a_4   a_3   a_2   a_1   a_0   0     0 
    0     a_4   a_3   a_2   a_1   a_0   0 
    0     0     a_4   a_3   a_2   a_1   a_0 
    4a_4  3a_3  2a_2  1a_1  0     0     0 
    0     4a_4  3a_3  2a_2  1a_1  0     0 
    0     0     4a_4  3a_3  2a_2  1a_1  0 
    0     0     0     4a_4  3a_3  2a_2  1a_1 

It's easy to see that there are no monomials in the expansion of this
involving either  a_4 * a_3 * a_2^4 * a_1  or  a_4 * a_2^6.

For larger n, it's not clear to me what restrictions need to be put on
the partitions to guarantee that the corresponding monomials occur in the
expansion of the determinant.  Columns near the left or right have very
few nonzero elements, and this adds some restrictions to the partitions.
For example, from column 2 of the matrix, we see that the partition must
have at least one term equal to n or n-1.  From the last column, it must
have at least one term equal to 0 or 1.  Maybe the complete list of such
conditions is enough; I don't know.

Even if we could figure out exactly which partitions correspond to
monomials that occur in the expansion, I can't rule out the possibility
that the coefficients of some such monomial could cancel out, further
reducing the number of nonzero monomials in the discriminant.

Dean Hickerson
dean at math.ucdavis.edu