"degenerate": I don't like the term, but it seems to
be in common use. A rank-N truth-table T (in the J sense:
shape N#2) is degenerate if it is logically equivalent
to a truth-table of rank < N, or if {.T on some axis is
identical to {:T.
10 as the number of nondegenerate tables of rank 2:
The number of possible rank-2 tables with
0 1 2 3 4 ones is
1 4 6 4 1
The tables with 0 or 4 ones are degenerate because
equivalent to 0 or to 1.
The tables with 1 or with 3 ones are nondegenerate.
Four of the tables with 2 ones have the ones either
on a a row or a column, so have {. T the same as {:T
on some axis. The two tables with the ones on diagonals
are nondegenerate.
So 4 + 4 + 2 nondegenerate.
"laminate": If it weren't for the scalar case, I'd
just use ",:". For A and B of the same shape (S, say)
$A,:B is 2,S; except where A and B are scalars, where
$A,:B is 2 1, and I want it to be just 2. So the clumsy
lam=: ,`,:@.(0 < #@$@])
Best wishes, N.
On 2019.08.18 08:55:06, you,
the extraordinary Raul Miller, spake thus:
> From: Raul Miller <[email protected]>
> Subject: Re: [Jchat] Number of self-dual arrays of rank N; OEIS
> Date: Sun, 18 Aug 2019 08:55:06 -0400
> To: [email protected]
>
> As stated here, I think your succinct statement leaves out too much.
>
> It probably needs "truth value" and something related to "laminated".
>
> I am also dubious of the number 10, in that sequence, as it's described
> here, as you left out the concept behind "degenerate".
>
> Thanks,
>
> --
> Raul
>
> On Sun, Aug 18, 2019 at 12:36 AM Nollaig MacKenzie <
> [email protected]> wrote:
>
> >
> > 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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> >
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