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     1. The weights of m-hppd's


We consider m-multinomial coefficients

æ
ç
è
   n
k1
k2
... 
km
ö
÷
ø
Let (a1, a2,...,am) be an m-vector of positive integers; and, for each l, 0 £ l £m, let 1£il £m, ir<is, if r<s and
V(i1,...,il) =  æ
ç
è
n    +    ai1
...
+ail      
k1¢
k2¢
... 
km¢
ö
÷
ø
      (1.1)
where
ì
í
î
kij¢ = kij+aij
j = 1,2,...,
ki¢ = ki
otherwise 
      (1.2)

Then {V(i1,...,il)} is the set of vertices of an m-hppd P in the Pascal m-simplex, with its edges parallel to the coordinate axes k1=constant, ..., km=constant. We define the weight of P, denoted W(P), by the formula

W(P) = 

Õ
even
V(i1,...,il)


Õ
odd
V(i1,...,il)

Notice that there are 2m-1 terms V(i1,...,il) with l even, and 2m-1 terms V(i1,...,il) with l odd. A straightforward calculation shows the following.

Proposition 1.1 W(P) is independent of k1, k2,..., km, and invariant under the action of the symmetric group Sm on the set {a1, a2,..., am}, given by pai = api,    p Î Sm.

The vertex

æ
ç
è
   n
k1
k2
... 
km
ö
÷
ø

given by l=0 in (1.1) is called the anchor of P. It is obviously invariant under the action of Sm. Let us now assume that the integers a1, a2,..., am are all distinct, and let p Î Sm act on {a1, a2,..., am} 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,

( Õ even     vertices     of     P)( Õ odd    vertices     of    P*) = 
( Õ odd     vertices     of     P)( Õ odd    vertices     of    P*)       (1.4)

Notice that there are 2m-1 vertices on each side of (1.4). These may be described as the 2m+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 {a1, a2,..., am} fixed, j ³ 1. Then P and P* share 2j common vertices (including the common anchor). Of these vertices 2j-1 are even and 2j-1 are odd. Thus, when we cancel out the common vertices, we get a statement like (1.4), but with 2m-2j vertices on each side. In fact, this statement is compounded of 2j 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 {a1, a2,..., am} 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 k1, k2,..., km as the distinguished variable. To simplify notation (as much as possible) we will deal explicitly with the case in which km is the distinguished variable.

We choose positive integers (a1, a2,..., am-1) with a £ n. Then, for each l, 0£l£m-1 let 0 < il < m, ir < is if r < s, and

V0(i1,...,il) =  æ
ç
ç
ç
ç
è
   
k1¢
... 
km-1¢

km

¢
ö
÷
÷
÷
÷
ø
      (1.5)
where
ì
í
î
kij¢ = kij+aij
1 £ j £l
ki¢ = ki,
otherwise 
      (1.6)
so that

km

= n- m-1
å
 i = 1
ki¢ = n- m-1
å
 i = 1
ki-  l
å
 j = 1
aij = km-  l
å
 j = 1
aij      (1.7) 

We say that V0, is an even vertex if l is even. This, of course, includes the case where l=0, when V0 is the anchor. If l is odd, V0 is an odd vertex.

Next let

Va(i1,...,il) =  æ
ç
ç
ç
ç
è
    n-
k1¢
... 
km-1¢

km

¢
ö
÷
÷
÷
÷
ø
      (1.8)
ì
í
î
kij¢ = kij+aij
1 £ j £
ki¢ = ki,
otherwise 
      (1.9)
so that

km

= n-a-  m-1
å
 i = 1
ki¢ = n-a- m-1
å
i = 1
ki- l
å
j = 1
aij = km-a-  l
å
j= 1
aij       (1.10) 

We say that Va is an even vertex if l is odd; and Va is an odd vertex if l is even.

The vertices V0(i1,...,il), Va(i1,...,il) constitute the 2m vertices of an m-hppd P and we define, as before, the weight of P by the formula

W(P) = 
Õ even     vertices     of     P

Õ odd     vertices     of     P
      (1.11)
Just as in Proposition 1.1 we have

Proposition 1.2 W(P) is independent of n, k1, k2,..., km-1 and invariant under the action of the symmetric group Sm on the set {a1,a2,...,am-1,a}, given by pai = api, pÎSm where we identify am 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 ai be positive, so long as we stay within the Pascal m-simplex.


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