From the experience with the left descent to 27 and its indication that 2^a*3^b*n+c is the form of the descriptors of the several members of the left descent, and the experience with a number of simpler left descents, via downward development and upward development, we can establish some general rules for the formation of the 2^a*3^b coefficients.
We call the coefficients x(i) where i runs from 1 for the extension element which heads the left descent to omega for the leaf element which closes the left descent. The coefficients all have both a non-zero power of 2 and a non-zero power of 3 as factors. Each rule below completely determines the power of the 2 or 3 it is concerned with -- there is no effect on the value from one rule by another rule. The resulting coefficients are those of left descents anchored at both ends.
(1) The x(1) coefficient has 2^3 as a factor.
(2) The x(1) coefficient has 3^omega as a factor.
(3) The exponent in the 2^a factor in the coefficients goes up by 2 for each b step and by 1 for each s step from x(1) to x(omega) as the descent is traversed downwards.
(4) The exponent in the 3^b factor in the coefficients goes down 1 in each step from x(1) to x(omega) as the descent is traversed downwards. (See the end of this page for a corollary.)
(5) The constant c is correctly calculated by Maple and/or by a process like that illustrated above.
Applying these rules in the esbt case, where omega is 3, we get:
coefficient of x(1) will be 2^3*3^3 or 216,
coefficient of x(2) will be 2^4*3^2 or 144, and
coefficient of x(3) will be 2^6*3^1 or 192,
in complete accord with the more arduously derived coefficients by the downward development and upward development.
There's a corollary, belatedly discovered from observation of Figure 9, the example illustrating the location of an arbitrary odd integer in the abstract predecessor tree. Comparison of the coefficients in columns 6 and 7 show that each generation sees an increase in their ratio by a factor of 3 from the bottom up. I.e., each step of the bottom-up subsetting process identifies subsets which are 1/3^n of the size of the entire node contents counting from the bottom up. This merely reflects the division into thirds which each step in the abstract predecessor tree development incurs.
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