I just filled a gap in my education by learning about the Cayley-Menger theorem, and the Cayley-Menger determinant:

If $P_0, \dots, P_n$ are $n+1$ point in $\mathbb{R}^n$, and $d_{i,j} = |P_i - P_j|$ is the Euclidean distance from $P_i$ to $P_j$, we first form the $n+1 \times n+1$ matrix of squares of the distances (say $B$) and then form an $n+2 \times n+2$ matrix $A$ which has $B$ in the lower right hand corner, and has all the elements in the first row and column $=1$ except for the one in the upper left hand corner, which is 0.

The Cayley-Menger theorem (which is an $n$ dimensional generalization of Heron's formula for the area of a triangle) says that if $V$ is the volume of simplex whose vertices are the $P_i$, then

$(-1)^{n-1} 2^n (n!)^2 V^2 = \det(A)$.

I was interested in the structure of $\det(A)$ as a polynomial. There is a nice paper "The Cayley-Menger Determinant is Irreducible for $n \ge 3$" by Carlos d'Andrea and Martin Sombra (arXiV:math/0406359). When I calculated the number of monomials in each of the polynomials obtained by evaluating $\det(A)$, I got the sequence (starting with $n=1$):

1,1,6,22,130,822, 6202, 52552

which is sequence A002137 in the OEIS: Number of n X n symmetric matrices with positive entries, trace 0 and all row sums 2. There's no mention there of the Cayley-Menger determinant.

So the question I have, is there a one-to-one correspondence that one can find between the matrices and monomials in $\det(A)$? Even nicer, would be to find the value for the coefficient of the monomial from the matrix.

Added Later: I should have looked at the reference in the OEIS: A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.

In it he gives a recurrence for the number terms in a symmetric matrix with 0's on the diagonal, and produces the same sequence. However, I still can't find a connection with the coefficients.