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From: OsherD on 1 May 2010 01:41
From Osher Doctorow

The "needed" 3-dimensional generalization of Catalan Numbers are Fuss-
Catalan Numbers, defined by J. - C. Aval of U. Bordeaux France (and
others) in "Multivariate Fuss-Catalan numbers," arXiv: 0711.0906 v1
[math.CO] 6 Nov 2007, 12 pages, by:

1) C_p(n) or Cp(n) = 1/[(p - 1)n + 1] C(pn, n) where C(pn, n) =
number of unordered combinations of n things chosen from pn things.

These for p = 3 count ternary trees, which are trees such that every
internal node has exactly 3 children, and have 3-dimensional
tetrahedron interpretations analogous to 2-dimension Catalan number
triangle interpretations and similarly for their corresponding arrays.

We have:

2) C3(n) = [1/(n + 1)] C(3n, 3)

It turns out that just as Catalan numbers are sums of "ballot numbers"
in 2 dimensions, so Fuss-Catalan numbers are sums of "ballot numbers"
in 3 dimensions, where the 2-dimensional ballot number B(n, k) is:

3) B(n, k) = [(n - k)/(n + k)]C(n + k, n)

and:

4) C(n) [nth Catalan number] = sum B(n, k) where sum is for k = 0 to
n - 1.

For 3-dimensional ballot numbers B3(n, k, L):

5) B3(1, 0, 0) = 1, B3(n, k, L) for n > 1 and k + L < n is sum B3(n -
1, i, j) where sum is over:
6) 0 < = i < = k, 0 < = j < = L

We have analogous to (4):

7) C3(n) = {1/[2n + 1]}C(3n, n) = sum B3(n, k, L) where sum is over k
and L, n integer < = 0.

The quantities B3(n, k, L) also have an explicit equation:

8) B3(n, k, L) = C(n+k, k)C(n+L-1, L) [n-k-1]/[n+k]

Notice in (8) the appearance of n - k - 1 which is -(1 + y - x) for y
= k, x = n, where 1 + y - x is the usual Probable Causation/Influence
provided that the integers are normalized (roughly speaking, divided
by a constant integer larger than all of them). This type of thing
occurs also in equation (1) in the denominator.

Osher Doctorow

From: OsherD on 1 May 2010 01:47
From Osher Doctorow

Regarding (5), I should mention also that B3(n, k, L) = 0 if k + L > =
n.

J. - C. Aval has 22 papers in arXiv, 2001 through 2009.

Osher Doctorow
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