Information Technology Reference
In-Depth Information
The condition for a monomial, given by the expansion in figures of the degree of each
of the variables, to be associated to an arborescent monomial is easily determined:
the largest index n must appear only once, and in head position, and all the others
i
,
n
2, must appear as many times in head position than in tail position.
This condition can be expressed by a boolean function of the degrees of low termic
complexity, taking the value 1when satisfied and 0 if not; let us call it the compatibility
condition.
To check that this condition is sufficient, we observe that, before placing the
variables of head
i
, we can place the variables of head bigger than
i
; this done,
we can add to the tree the variables of head
i
, since the number of tail position
corresponds, and then the variables of tail
i
>
i
∈
1, etc.
For instance, in the example above there is only one second arborescent monomial
giving the same monomial (therefore the coefficent of this monomial is 2), the only
degree of freedom being the choice for the placement of the two variables with head
3. This second arborescent monomial is:
−
310
540
→
432
210
654
321
→
310
432
210
To compute the coefficient-function of
P
n
, we must count how many arborescent
monomials are produced by a given monomial. If we note
C
i
the number of ways to
place the variables of head
i
once the variables of bigger head are placed, number
that is independent of the anterior (and posterior) placements, then the coefficient of
