We consider m-multinomial coefficients
Then {V(i_{1},...,i_{l})} is the set of vertices of an m-hppd P in the Pascal m-simplex, with its edges parallel to the coordinate axes k_{1}=constant, ..., k_{m}=constant. We define the weight of P, denoted W(P), by the formula
Notice that there are 2^{m-1} terms V(i_{1},...,i_{l}) with l even, and 2^{m-1} terms V(i_{1},...,i_{l}) with l odd. A straightforward calculation shows the following.
given by l=0 in (1.1) is called the anchor of P. It is obviously invariant under the action of S_{m}. Let us now assume that the integers a_{1}, a_{2},..., a_{m} are all distinct, and let p Î S_{m} act on {a_{1}, a_{2},..., a_{m}} without fixed point. Then no other vertex of P is invariant under p. The permutation p. maps P to another m-hppd P^{*} which shares with P just their common anchor. Thus we obtain, by equating W(P) with W(P^{*}), a rule declaring that, omitting the anchor,
Notice that there are 2^{m}-1 vertices on each side of (1.4). These may be described as the 2^{m+1}-2 vertices of an m-hyperstar (of David), so (1.4) enunciates an m-Hyperstar of David Theorem. Suppose instead that the permutation p leaves j elements of the set {a_{1}, a_{2},..., a_{m}} fixed, j ³ 1. Then P and P^{*} share 2^{j} common vertices (including the common anchor). Of these vertices 2^{j-1} are even and 2^{j-1} are odd. Thus, when we cancel out the common vertices, we get a statement like (1.4), but with 2^{m}-2^{j} vertices on each side. In fact, this statement is compounded of 2^{j} cases of (1.4), each exhibiting an instance of the (m-j)-Hyperstar of David Theorem. In fact, we obtain a non-degenerate Hyperstar of David identity precisely when p is an m-cycle. When p is a composition of cycles on disjoint subsets of {a_{1}, a_{2},..., a_{m}} the resulting identity will always break down into a set of identities in lower dimensions. As an example, consider the case m=4 and the permutation p = (12)(34). Then the Hyperstar of David identity asserts the equality of two products, each involving 15 multinomial coefficients. However this identity is really composed of two genuine Star of David identities, each asserting the equality of two products of 3 coefficients, and a new identity asserting the equality of two products of 9 coefficients. It is natural to ask: "What is the geometric, and algebraic, meaning of this last equality?" We propose to study this type of question in a later article. So far we have treated n as a distinguished variable in our anchor formula. It is, of course, evidently possible to treat any of k_{1}, k_{2},..., k_{m} as the distinguished variable. To simplify notation (as much as possible) we will deal explicitly with the case in which k_{m} is the distinguished variable. We choose positive integers (a_{1}, a_{2},..., a_{m}_{-1}) with a £ n. Then, for each l, 0£l£m-1 let 0 < i_{l} < m, i_{r} < i_{s} if r < s, and
We say that V_{0}, is an even vertex if l is even. This, of course, includes the case where l=0, when V_{0} is the anchor. If l is odd, V_{0} is an odd vertex. Next let
We say that V_{a} is an even vertex if l is odd; and V_{a} is an odd vertex if l is even. The vertices V_{0}(i_{1},...,i_{l}), V_{a}(i_{1},...,i_{l}) constitute the 2^{m} vertices of an m-hppd P and we define, as before, the weight of P by the formula
Proposition 1.2 W(P) is independent of n, k_{1}, k_{2},..., k_{m-1} and invariant under the action of the symmetric group S_{m} on the set {a_{1},a_{2},...,a_{m}_{-1},a}, given by pa_{i} = a_{pi}, pÎS_{m} where we identify a_{m} with a. The reader may now supply the analogues of the remarks above leading to an m-Hyperstar of David Theorem. Of course we do not need to insist that the a_{i} be positive, so long as we stay within the Pascal m-simplex. |