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 (n+1), the multinomial coefficients

at level n we have the m(m-1) multinomial coefficients

and at level (n-1) we have the m multinomial coefficients

These m(m+1) coefficients, as they prove, may then be partitioned into m sets, each containing (m+1) coefficients, in such a way that the product of the coefficients in each set is a given number N, namely

which is thereby seen to be an integer. The method of partitioning is not claimed to be unique. This theorem may, in the case m=2, be regarded as a restatement of the Star of David Theorem, in the case a₁=1, a₂=-1; notice that we may certainly allow negative integers among the a_i in the Star of David Theorem, so long as k_j+a_i remains positive. However, for 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.

Theorem 3.1 Given the multinomial anchor

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 n, k₁, k₂,...,k_m is that it be a separable function; thus we may replace the multinomial coefficient by a function

An example of such a separable function is the q-analogue of the multinomial coefficient. This is the Gaussian polynomial (see [7]), obtainable from the multinomial coefficient

may be regarded as the coefficient of a₁^k₁a₂^k₂...a_m^k_m in the expansion of (a₁+a₂+...+a_m)ⁿ in the algebra over R generated by (a₁,a₂,...,a_m; q) subject to a_ja_i=qa_ia_j, i<j, qa_i=a_iq. Notice that, in this generalization of the material of Section 1, it is not necessary that the functions f₁, f₂,...,f_m be the same; however, if 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_m, then the functions f₁, f₂,...,f_m may be chosen to be the same, given that F is separable.

The material of Section 2 makes it obvious how we would extend the definition of F, given by (3.3) to the domain of arbitrary integers n, k₁, k₂,...,k_m. The formula (2.5) is easily generalized by replacing the multinomial coefficient by 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 F be invariant under the action of S_m. We leave the details to the reader.