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From: OsherD on 1 Jun 2010 01:57
From Osher Doctorow

With the qualification again that the author of the paper to be cited
does not make explicit mention of Probable Causation/Influence (PI),
David Callan of U. Wisconsin-Madison USA has a new paper:

1) "Flexagons yield a curious Catalan number identity," arXiv:
1005.5736 v1 [math.CO] 31 May 2010, 5 pages.

Flexagons are roughly speaking flat plane figures that consist of
folded paper which when unfolded reveal faces not visible when
folded. They include hexaflexagons (formed from a hexagon).

Flexagons are abstractly representably by recursively defined
permutations called PATS, which are counted by Catalan numbers, and
counting them in a particular way by using the number of "descents"
gives the identity:

2) C_n = sum [1/(2n - 2k + 1)] C(2n - 2k + 1, k)C(2k, n-k), sum from k
= 0 to n.

where Cn = the Catalan number = (1/(n+1))C(2n, n) and C(a, b) = a!/[(a-
b)!b!] where the factorial for example of 5 is 5 times 4 times 3
times 2 times 1.

Callan proves that this is distributed (the number of descents on
pats) as even level vertices in binary trees. Similar consideration
of ternary and higher order trees yields the same equation as equation
(2) with 2 replaced by r everywhere with r > = 2.

Readers will notice that:

3) 2n - 2k + 1 = P(2k --> 2n) (the Probable Causation/Influence (PI)
of 2k on 2n)

and similarly for rn - rk + 1 = P(rk --> rn), if normalized.

Osher Doctorow
From: OsherD on 1 Jun 2010 02:03
From Osher Doctorow

A DESCENT in a permutation p is a pair of adjacent entries (pi, pi+1)
such that pi > pi+1, where i, i+1 are subscripts.

A PAT is defined recursively as first a singleton, then a PAT of
length > = 2is a PAT iff there is a unique split point that divides p
into subpermutations p1, p2 with all entries in p1 > all entries in p2
and also the reverse of p1 and p2 is a PAT.

Osher Doctorow
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