533]
FACTORIALS, AND DERIVATIONS.
471
this is evolved from the top term (a, a) by a process given implicitly in Arbogast’s
Calculus of Derivations, and which may be termed the rule of the last and the last
but one. Let the direction “operate on any letter,” be understood to mean that the
letter in question is to be changed into that which immediately follows it, but in
such wise that when the letter occurs more than once, e.g. as in a, a the operation
affects only the letter in the right-hand place. Then operate on the a, a in regard
to a, we obtain a, ¡3; operate on this in regard to /3, we obtain a, 7; and in regard
to a, we obtain /3, /3. Again we operate on a, 7 in regard to 7, and obtain a, 8; we
do not operate on it in regard to a for the reason that a is not the letter immediately
preceding 7. Operate on ¡3, 0 in regard to /3, we obtain /3, 7. The next step, if the
series extended to e would be to operate on a, 8 in regard to 8, giving a, e; do not operate
on it in regard to a, for the reason that a is not the letter immediately preceding 8.
But in the example, since the series does not extend to e, there is no operation on
a, 8. Passing then to the next term /3, 7, we operate in regard to 7, obtaining /3, 8,
and since ¡3 is the letter immediately preceding 7, we also operate in regard to ¡3,
obtaining 7, 7. Similarly if e were admissible, /3, 8 would give /3, e, but it in fact
gives nothing; 7, 7 gives 7, 8; thus if e were admissible would give 7, e and 8, 8,
but it in fact gives only 8, 8, and, e being inadmissible, the process is here concluded.
The rule is, operate on the last letter, and when the last but one letter is that
which, in alphabetical order, immediately precedes the last letter (but in this case
only) operate on the last but one letter.
Taking another example, but with numbers instead of letters, and supposing the
highest admissible number to be 5, then from 111 we derive as follows:
113
114
115
125
135
145
155
255
122
123
124
134
144
235
245
345
222
133
224
225
244
335
444
223
233
234
334
344
333,
too OOO
the original single column being here for greater convenience broken up into distinct
columns ; but the order of the terms, when the columns are taken one after the other
in order, each being read from the top to the bottom, being the same as before; it
will be noticed that the successive divisions are the partitions into 3 parts (no part
exceeding 5) of the numbers 3, 4, ..., 15 respectively; the partitions being in each
case obtained without repetition, and those of the same number being given, say in
their numerical order (corresponding with the alphabetical order when letters are
employed). It is necessary to show that the partitions will be obtained without
repetitions; and that all the partitions will be obtained; for this purpose consider, for
example, the partitions of 9 ; any one of these is either a partition 135 where the last
number 5 is not a repeated number; and in this case there is a partition of 8, viz.
134, from which operating on the last we obtain 135 ; but there is no other partition
of 8 which would give 135, the only such partition would be 125, but here, as 2 is
not the number which immediately precedes 5, there is no operation on the last but
one, and we do not from it obtain 135. Or else a partition of 9 is of the form 144