[seqfan] Re: Organization of dispersion-arrays.

Antti Karttunen antti.karttunen at gmail.com
Tue Dec 15 17:34:32 CET 2015

On Tue, Dec 15, 2015 at 6:14 PM, Frank Adams-Watters <franktaw at netscape.net>

> 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.

[A bit tired now, so I might make a mistake here, but ...]: I have assumed
that when you write about two sequences that are mutually ordinal
transforms of each other, then it is question of the row and column indices
of such arrays. So it is actually the point
 (B) "The column index and the row index" in my mail, what we are talking
about here, not the point (C) quoted below?

So for example, if I understand your terminology right, sequences
and https://oeis.org/A254112 are ordinal transforms of each other?

Best regards,


> 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.

Yes, certainly.

> 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.

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