I was a little dissatisfied with the discussion, in my “A
NOTE ON TRUTH-FUNCTIONAL SELF-DUALITY” in the June 2018
Journal of J, of the number of self-dual N axis arrays
(N= 0,1,2,….). A succinct statement is: The number of
self-dual arrays of N axes = the number of non-self-dual
non-degenerate arrays of N-1 axes.
This follows from the fact that the {. and {: of a rank
N self-dual must be duals, one of the other. So each of
the nondegenerate N -1 rank arrays gives rise to a rank N
self-dual, except for the N-1 rank self-duals, which if
laminated with their duals would yield an N rank array with
{. and {: identical.
I use “laminate” a little idiosyncratically here. I mean
lam=: ,`,:@.(0<#@$@])
To illustrate: There are 2 non-self-dual non-degenerate
arrays for N=0, namely scalar 0 and 1.
So there will be 2 self-dual arrays for N=1, namely
0 lam 1 1 lam 0
Thus there are 0 non-self-dual nondegenerate arrays of rank
1, and therefore no self-duals of rank 2.
There are 10 non-self-dual non-degenerate arrays of rank 2,
and therefore 10 self-duals of rank 3.
The sequence is: 0,2,0,10,208….
By this reasoning the next term (for N=5) should be 64386. I
went to OEIS.org to check, and found that 0,2,0,10,208
wasn’t listed.
I’m tempted to submit the sequence to OEIS, but I
don’t have an elegant formula for generating it, and
I don’t know whether it is of any significance. As a
non-mathematician I don’t want to make a fool of myself.
Any mathematical types here have advice (other than
“nothing wrong with making a fool of yourself”)?
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