[seqfan] Re: Organization of dispersion-arrays.

[Frank Adams-Watters]

If I understand this correctly, what you are here calling the "complementary" sequence is what I have been calling the "ordinal transform" of the sequence. I often find it easier to think of the related sequences as transforms of each other rather than by looking at the associative array.

Whatever sequence you feed to the ordinal transform - even a non-mathematical sequence such as a sequence of words - the result will be a sequence of positive integers with the property that for any k and n > 0, in the first n terms of the sequence, k will occur at least as many times as k+1. This property is characteristic of this transform, since if you apply the transform twice to a sequence with this property, you get the original sequence back. This is easy to see in terms of the array: you are just transposing it.

If the original sequence is fractal - that is, invariant under an upper trim: removing the first occurrence of each value in the sequence gives you back the same sequence - then its ordinal transform will be invariant under the lower trim: remove the 1's from the sequence, and subtract 1 from every remaining term; and vice versa, of course.

I guess my main point is that, when complementary sequences are in the database, I would like to see a note that they are ordinal transforms of each other, whether in a comment, cross-ref, or formula.

Franklin T. Adams-Watters


-----Original Message-----
From: Antti Karttunen <antti.karttunen at gmail.com>
To: Sequence Fanatics <seqfan at list.seqfan.eu>
Sent: Tue, Dec 15, 2015 2:22 am
Subject: [seqfan] Organization of dispersion-arrays.

C) First few rows and first few columns. Depending on the orientation of
the array (which of two transposes it is), the topmost row or the leftmost
column is the complementary sequence to the sequence whose dispersion this
array is, and correspondingly, the edgemost sequence on other axis gives
the iterates of the dispersion sequence itself.