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A229874 An enumeration of all sorted k-tuples containing positive integers.

%I A229874 #9 Oct 06 2013 14:45:23
%S A229874 1,2,1,1,3,2,1,2,2,1,1,1,4,3,1,3,2,3,3,2,1,1,2,2,1,2,2,2,1,1,1,1,5,4,
%T A229874 1,4,2,4,3,4,4,3,1,1,3,2,1,3,2,2,3,3,1,3,3,2,3,3,3,2,1,1,1,2,2,1,1,2,
%U A229874 2,2,1,2,2,2,2,1,1,1,1,1,6
%N A229874 An enumeration of all sorted k-tuples containing positive integers.
%C A229874 Begin with the 1-tuple (1), and then reading from the beginning of the list of k-tuples append to the list (n+1) if the k-tuple read is a 1-tuple and for all cases, append the (k+1)-tuples (...,n,1), (...,n,2), ..., (...,n,n), where n is the last element of the k-tuple that was read.
%C A229874 This sequence is a flattening of that process.
%C A229874 Each tuple contains a unique group of integers, meaning that the sequence of tuples is an enumeration of all finite sets of positive integers.
%C A229874 Determining a tuple's parent is as simple as removing the last element in the case of k-tuples where k>2 and by subtracting 1 from the only element in the case of 1-tuples. E.g., (7,5,3,2,1)'s ancestry is (7,5,3,2), (7,5,3), (7,5), (7), (6), (5), (4), (3), (2), (1).
%C A229874 Tuples are in ordered so that the rightmost element increases in value from sibling to sibling, resembling place-value notation. This has the side effect of putting the values within the tuples in the reverse of the usual sort order. The alternative version of this sequence with tuple values in increasing order can be found in A229897.
%C A229874 Remarkably, the k-tuple sizes can be found in A124736 - k repeated C(n,k-1) times - and relatedly, the first appearance of n in this sequence is at position 2^(n-1)+1.
%H A229874 Carl R. White, <a href="/A229874/b229874.txt">Table of n, a(n) for n = 1..1025</a>
%H A229874 Carl R. White, <a href="/A229874/a229874.txt">Tabular layout of the sequence showing the k-tuples as they occur</a>
%e A229874 Sequence begins (1), (2), (1,1), (3), (2,1), (2,2), (1,1,1), (4), etc.
%Y A229874 Cf. A001057. All tuples, not just sorted: A229873. Alternative version: A229897.
%K A229874 nonn,tabf
%O A229874 1,2
%A A229874 _Carl R. White_, Oct 02 2013