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A229874 An enumeration of all sorted k-tuples containing positive integers. 3
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, 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, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

This sequence is a flattening of that process.

Each tuple contains a unique group of integers, meaning that the sequence of tuples is an enumeration of all finite sets of positive integers.

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).

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.

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.

LINKS

Carl R. White, Table of n, a(n) for n = 1..1025

Carl R. White, Tabular layout of the sequence showing the k-tuples as they occur

EXAMPLE

Sequence begins (1), (2), (1,1), (3), (2,1), (2,2), (1,1,1), (4), etc.

CROSSREFS

Cf. A001057. All tuples, not just sorted: A229873. Alternative version: A229897.

Sequence in context: A273135 A165162 A125106 * A330439 A243611 A273102

Adjacent sequences:  A229871 A229872 A229873 * A229875 A229876 A229877

KEYWORD

nonn,tabf

AUTHOR

Carl R. White, Oct 02 2013

STATUS

approved

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)