Sequences containing all finite sequences

[Robert G. Wilson v]

%I A000001
%S A000001 1,1,1,2,1,1,1,1,2,2,1,3,1,1,1,1,1,1,2,1,2,1,2,1,1,1,3,2,2,3,1,4,1,1,1,
%T A000001 1,1,1,1,1,2,1,1,2,1,1,2,1,1,2,1,1,1,1,1,3,1,2,2,2,1,2,1,3,1,2,2,1,3,1,
%T A000001 1,1,4,2,3,3,2,4,1,5,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1
%N A000001 The permutations of all partitions of n in lexicographical order.
%t A000001 Flatten[ Table[ Reverse[ # ] & /@ Reverse[ Sort[ Flatten[ Permutations[ 
# ] & /@ Partitions[ n], 1]]], {n, 6}]] (from RGWv (rgwv at rgwv.com), Jun 22 2005)
%Y A000001 Cf. .
%O A000001 1,4
%K A000001 nonn,tabl
%A A000001


{1}
{1, 1}, {2}
{1, 1, 1}, {1, 2}, {2, 1}, {3}
{1, 1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {2, 1, 1}, {1, 3}, {2, 2}, {3, 1}, {4}
{1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 2, 1}, {1, 2, 1, 1}, {2, 1, 1, 1}, {1, 1, 
3}, {1, 2, 2}, {2, 1, 2}, {1, 3, 1}, {2, 2, 1}, {3, 1, 1}, {1, 4}, {2, 3}, {3, 2}, 
{4, 1}, {5}
{1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 2}, {1, 1, 1, 2, 1}, {1, 1, 2, 1, 1}, {1, 2, 1, 
1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 2, 1, 2}, {2, 1, 1, 2}, 
{1, 1, 3, 1}, {1, 2, 2, 1}, {2, 1, 2, 1}, {1, 3, 1, 1}, {2, 2, 1, 1}, {3, 1, 1, 
1}, {1, 1, 4}, {1, 2, 3}, {2, 1, 3}, {1, 3, 2}, {2, 2, 2}, {3, 1, 2}, {1, 4, 1}, 
{2, 3, 1}, {3, 2, 1}, {4, 1, 1}, {1, 5}, {2, 4}, {3, 3}, {4, 2}, {5, 1}, {6}



hv at crypt.org wrote:

> franktaw at netscape.net (Franklin T. Adams-Watters) wrote:
> :Which sequences contain all finite sequences of non-negative integers as subsequences?  (One can also add 1, and look for sequences containing just all finite sequences of positive integers as subsequences.  However, the natural examples I found all involved non-negative integers, so that's the way I'm framing the problem.)
> 
> This is a nice concept.
> 
> One sequence that maps more readily to the positive integers is the
> concatenated list of all compositions of successive n:
> 
> 1,
> 1,1, 2,
> 1,1,1, 1,2, 2,1, 3,
> 1,1,1,1, 1,1,2, 1,2,1, 1,3, 2,1,1, 2,2, 3,1, 4,
> etc (not in OEIS, but cf A045623)
> 
> :Are there other sequences with this property in the OEIS - or that should be in the OEIS?  Note that all 3 of these are tables, where every finite sequence occurs in a row in an obvious way.  It would be nice to find sequences not defined in this way.
> 
> I'd expect all such sequences to grow very slowly, which may reduce their
> usefulness within the OEIS other than when viewed particularly as examples
> of this property.
> 
> It looks like A066099 may also be a positive integer example, though
> possibly just a minor variation of your factorisation example;
> I stumbled across it checking for presence of the above.
> 
> Hugo
>