# Sequences containing all finite sequences

Franklin T. Adams-Watters franktaw at netscape.net
Tue Jun 28 02:19:34 CEST 2005

Robert's sequence:

"Robert G. Wilson v" <rgwv at rgwv.com> wrote:
>%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

is very similar to
>Second is the sequence of with number of zeros following each 1 in the
>binary representation of n.  This is probably the easiest one-to-one
>correspondence between non-negative integers and finite sequences of
>non-negative integers.  It starts:
>0
>1
>0,0
>2
>1,0
>0,1
>0,0,0
>3
(Which is now A108730.)  My values are one smaller, and the lines in each group are sorted in reverse lexicographical order.  Incidently, the incremented values are in A066099, with an initial zero.  The name of that sequence calls it the sequence of exponents of A057335; it would be better to move that to the comments, and describe it as the sequence of combinatorial compositions.  (Really, A057335 is derived from this sequence, not vice versa.)

hv at crypt.org wrote:
>I think "combinatorial composition" is the correct term, at least
>judging by http://mathworld.wolfram.com/Composition.html.
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