[seqfan] Re: Sequences for Orderings
franktaw at netscape.net
franktaw at netscape.net
Fri Mar 26 23:29:32 CET 2010
Note that, for the reverse ordering, n [<] m iff n > m, the index
sequence is the all ones sequence, A000012.
I believe that the index sequence for Sharkovsky's ordering starts
1,1,1,2,2,3,3,5,4,6; this is not in the OEIS.
I think this is a good idea, but it needs to be documented; someone
going to mathematical literature or on-line reference sources won't
find it. Basically, a quick description like that in Marc's note
(quoted below) should either be included in a sequence (the index
sequence for Sharkovsky's ordering seems like a good choice) or added
to the OEIS as a separate document. Whichever is done, this can then
be referenced anywhere the idea is used.
Just looking at finite sequences, there are a number of important
sequences that can be enumerated by counting restrictions of this idea.
With no further restriction, we are counting the factorial numbers
A000142. If we require a(n+1) <= a(n) + 1, we get the Catalan numbers,
A000108. Requiring a(n+1) <= max(a(1),...,a(n)) + 1 gives us the Bell
numbers A000110 (and this a standard representation for set
partitions). Restricting so that the number of occurrences of each m
in a(1)..a(n) is at least the number of occurrences of m+1 in the same
sequence gives us the involutions, A000085 (these are trivially in
one-to-one correspondence with the Young tableaux). Requiring a(n) <=
a(n+1) <= a(n) + 1 gives us the count of compositions, A011782
(essentially powers of 2). Apply both of these last two restrictions
and we are counting partitions, A000041.
Franklin T. Adams-Watters
From: Marc LeBrun <mlb at well.com>
The OEIS contains many sequences which are "permutations of the
Naturally each such permutation a(n) presents an "ordering of the
determined by what values appear earlier in the sequence.
Let's write the ordering relation as [<] ("bracket-less-than"), defined
a(j) [<] a(k) just when j < k
(and of course similarly for [<=], [>], [>=], [min], [max] and so on).
An example permutation sequence is A065190: 1, 3, 2, 5, 4, 7, 6, 9,
wherein the implied ordering relation gives, for instance, that 3 [<] 2
because 3 appears before 2.
However, as discussed here previously, there are many interesting
of the integers which are NOT permutations, a favorite example being the
Alas it can't ever be directly entered in the OEIS, the way a
can, even in principle, because no matter how large a B-file we gave it
would be indistinguishable from just the sequence of odd numbers >= 3.
One way to address this would be a table where T(j,k) encodes when j
However I'd like to instead propose that a nicer way to represent an
arbitrary ordering relation [<] is as a sequence where a(n) gives the
of n when [1, 2, 3,... n] is sorted with respect to [<].
First, all permutation sequences can be encoded this way. For example
A065190 is represented as 1, 2, 2, 4, 4, 6, 6,... = A131005
Moreover this works for any ordering. For example to represent the
ordering "all odds < all evens" we get (an offset of) the badly named
obvious mixture of two sequences" A029578 = 1, 2, 2, 4, 3, 6, 4, 8, 5,
In this scheme every ordering is represented by what I call a "gradual"
sequence, where 1 <= a(n) <= n, and every such sequence corresponds
to an ordering (eg the "all 1's" sequence A000012 represents the
"reverse order", that is, sorted by [<] := >!)
Questions/suggestions/projects for Seqfans:
* Is this a good "canonical" scheme for entering orderings in the
* Is there an accepted name for the "index sequence" representing an
ordering? For the "gradual" property?
* Anybody want to submit the Sharkovsky ordering? Others?
* How about annotating all those "permutation" sequences with links
their index sequences, and vice versa? And perhaps even submitting the
missing partners for any really significant sequences?
More information about the SeqFan