Moments and frequencies
A distribution of data is characterized by its collection of moments as well as by the individual frequencies with which the data occur; these are really two complementary aspects of the same data set. Here, frequencies are associated with the nonnegative integers i, namely the number of configurations with a given number of ancestors. The moment of this data is defined by
where it is understood that .
The second volume, second edition of William Feller's popular book on the theory of probability  contains some material on moments; an advanced theory can be found in the American Mathematical Society Survey written by J. A. Shohat and J. D. Tamarkin .
Regarding the right hand side of Eq. 18 as the inner product of two vectors, then displaying the equations for individual moments, reveals a matrix equation
expressing the moments in terms of the frequencies. One desires a reversed relation, expressing frequencies using moments, the more so supposing that the moments could be calculated independently. Naturally this requires the determination of as many moments as frequencies, in order to produce a square matrix for inversion.
The matrix shown is a particular case of a type which is very well known in mathematics, the Vandermonde matrix. Its general form is
readily invertible through the use of the Lagrange interpolation polynomials , defined by
The inverse matrix transforms moments into frequencies: 0.30em
It is important to appreciate the mechanism by which the product yields the identity matrix. There are r+1 interpolation polynomials to evaluate at r+1 points; each polynomial vanishes at all the ``wrong'' points, while assuming unit value at the ``right'' point, namely the one for which the subscripts match. The matrix contains rows of coefficients for such polynomials; in the matrix product they form inner products with the columns of V, composed of all the powers needed to evaluate the polynomial at some particular point. The resultant interplay of right and wrong creates the unit matrix.
Because the ancestral data is all evenly spaced, the interpolation polynomials specialize to the Newton polynomials ,
whose characteristics have been nicely described, for example in Berezin and Zhidkov's numerical analysis text . The primary polynomial is symmetric or antisymmetric with respect to according to its parity, oscillating with diminishing amplitude from the origin to the midpoint, and growing rapidly outside the interval .
The coefficient in Eq. 21 is a Stirling number of the first kind, as described in Abramowitz and Stegun's mathematical handbook . The explicit formula cited therein is fairly long; asymptotic formulas, some special values and tables are also presented.
Since we have already seen how to express the moments of the counterimage distribution in terms of tensor powers of the de Bruijn fragments, we have an alternative viewpoint from which to examine the frequency of counterimages.
An important special case is that of zero variance, in which all data equal x, say; in that case one would have , for which the inversion formula Eq. 20 would yield . Supposedly x would be an integer, so that x's actual frequency r+1 would be recovered, all other frequencies vanishing. Nevertheless the formula is valid for any sequence of moments which are powers of x, such as might arise from an eigenvalue of one of the tensor power sums.
In practice, the high moments of any distribution are dominated by the largest datum; accordingly we might expect the inversion of a moment distribution to involve the maximal frequency plus a tail composed of the lower moments weighted by coefficients from the upper left hand corner of the matrix . These coefficients approach limiting values for large values of r, which would be the case of principal interest; moreover they diminish factorially while the moments increase as powers, ensuring convergence. For example, combining Equations 19 and 21 yields the frequency corresponding to the Garden of Eden,
We have just seen that vanishes (no Garden of Eden) for distributions with zero variance; the converse would be that it only vanishes for zero variance, a conclusion whose likelihood is greatly enhanced by the existence and appearance of Eq. 22.
Harold V. McIntosh