In [6] Hoggatt and Alexanderson give an interesting algorithm relating to the multinomial coefficients. We again take an anchor
and consider its nearest neighbours. Precisely we have, at level (
at level
and at level (
These
which is thereby seen to be an integer. The method of partitioning is not
claimed to be unique. This theorem may, in the case k+_{j}a remains positive. However, for
_{i}m³3,
the Hoggatt-Alexanderson Theorem diverges from what we have called the Hyperstar
of David Theorem.
On the other hand we may generalize the Hoggatt-Alexanderson Theorem in the following direction. We regard the theorem we have described as the case (1,-1) of the following theorem.
p,q)-satellites to consist
at level ( n+p) of the m coefficients
at level (
and at level (
p,q)
or (q,p).
Then these
N(p,q) is an integer.
For
Note that Of course the partitioning function is exactly that used in [6]; theirs is the hard work!
We may, just as for binomial coefficients, generalize the content of
Section 1
substantially beyond the domain of multinomial coefficients. Indeed, in
Section 1, all we require of the basic function of
separable
function; thus we may replace the multinomial coefficient by a function
An example of such a separable function is the
r! by
_{q} is just 1. The Gaussian polynomial
may be regarded as the coefficient of
a_{1}+a_{2}+...+a)_{m}^{n}
in the algebra over R generated by
(a_{1},a_{2},...,a; _{m}q)
subject to
a=_{j}a_{i}qa, _{i}a_{j}i<j,
qa=_{i}a. Notice that, in this
generalization of the material of Section 1, it is not necessary that the
functions _{i}qf_{1}, f_{2},...,f
be the same; however, if _{m}F in (3.3) is (as in the special case of the multinomial
coefficients and in the example of the Gaussian polynomials) invariant under
the action of the symmetric group S, then the functions
_{m}f_{1}, f_{2},...,f
may be chosen to be the same, given that _{m}F is
separable.
The material of Section 2 makes it obvious how we would extend the
definition of F; indeed, the generalization does not even require that F be
separable. However, one has to be careful in assigning a value to F
in the regions in which the multinomial coefficient takes the value 0;
this problem was first encountered in extending the harmonic triangle in
[4]. We do not want to put any stress on this technical point
here, since it is in the regions where formula (2.5) holds that the
real interest lies.
Finally, we may obviously generalize Theorem 3.1, provided that we insist in
(3.3) that |