x ¼ 19=32
w 4 ¼ 0:65605
w 0 ¼ 5:28125 w 5 ¼ 0:38950
w 1 ¼ 3:13574 w 6 ¼ 0:23133
w 2 ¼ 1:86815 :
w 3 ¼ 1:10493 :
Substituting here for the w
0 s the nearest integers, we obtain
w 0 ¼ 5, w 1 ¼ 3, w 2 ¼ 2, w 3 ¼ 1, w 4 ¼ 1, w 5 ¼ w 6 . . . 0,
Already from the fact that w 0 + w 1 + . . . should¼13, it is seen that again one of the
w
0 s must be increased by one unit. From those w
0 s that are set¼0, w 5 differs least
from the next highest integer. We want therefore w 5 ¼ 1, and obtain the complexion
0000011122345
whose digit sum is in fact¼19. The number of permutations this complexion is
capable of is
13!
5!3!2!
¼
13!
4!3!2!
Á
1
5
A complexion whose sum of digits is also¼19, and which one might suppose is
capable of very many permutations, would be the following:
0000111222334:
The number of permutations is
13!
4!3!3!2!
¼
13!
4!3!2!
Á
1
6
This is less than the number of permutations of the first complexion we found
from the approximate formula. Likewise, we expect that the number of permutations
of the two complexions
0000111122335
and
0000111122344
is smaller still. This is, for both complexions
4.3 Evolution of Thermodynamic State Index (Φ)
151