Determinants of Vandermonde -type matrix with mixed powers A certain combinatorics problem requires finding the determinant of matrices like this:
\begin{equation}
 \begin{bmatrix}
a & b & c & d & f \\
a^2 & b^2 & c^2 & d^2 & f^2 \\
a^m & b^m & c^m & d^m & f^m \\
a^{m+1} & b^{m+1} & c^{m+1} & d^{m+1} & f^{m+1} \\
a^{m+2} & b^{m+2} & c^{m+2} & d^{m+2} & f^{m+2} 
\end{bmatrix}  
\end{equation}
For each column, is an arithmetic progression of powers and then another arithmetic progression for ${m+y}$ for integer $m$
What is this type of matrix called? I cannot find any information about matrices of this form or a book or paper that describes how to find determinants  of this type of matrix.
In the above example, the has $C(5,3)=10$ permutations of triples of $a^m,b^m,c^m,d^m,f^m $ and then polynomials of $a,b,c,d,f$ associated with each in order to construct the determinant. One is $(a-b)(c-d)(c-f)(d-f)(cdf)^{m}$
 A: It does not have an elegant solution
Too many partitions. The reason  why the original  vandermonde matrix  has a nice form is because there is one partition regardless of the size.
So for the 7x7 version of the above matrix, we have 4 parts which have powers of m and 3 which do not. So the determinant will be of the form in which each 4-permutation of the 7 variables, for 35 different products.
Consider that  $a,b,c \approx z^{1/3}$ and $d,f,g,h \approx z$ as expansion of infinite series
So the determinant will be of the form $f_1(z)z^{4m}+f_2(z)z^{10m/3}+f_3(z)z^{8m/3}+f_4(z)z^{2m}$
So we have $C(4,1)C(3,3)+C(4,4)+C(4,3)C(3,1)+C(3,2)C(4,2)=C(7,4)=35$
$f_1(z)z^{4m}$ is the easiest and only has a single permutation, that being $C(4,4)$. This means that the $a^{m+y},b^{m+y},c^{m+y}$ terms can be zeroed-out, so you have a partition block matrix with one of the blocks zero, and determinant is easy to evaluate on this being the product of two small regular vandermonde matrices determinants, which is the product of 9 pairs (three for the 3x3 matrix and six for the 4x4 one). This can allow one to determine the asymptotic behavior of such matrices.
The next one has 12 permutations, so the determinant gets really messy fast.  This require zero-ing out triples of a,b...h and then swapping rows to make additional block matrices.
