cp's OEIS Frontend

Number k of elements to read from A229873 to obtain the next k-tuple.

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.

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

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 (1,n,...), (2,n,...), ..., (n,n,...), where n is the first element of the k-tuple that was read.

This sequence is a flattening of that process.

Other properties of this sequence are as A229874.

The two ones at the start of the parent sequence represent parent and child 1-tuples in the grandparent sequence [(1) and (2) respectively], hence this sequence also starts with 1, 1 rather than 2, which would otherwise be a more sensible way to describe the pair of ones.

All other elements are effectively run-lengths of strings of the same integer in A229895.

The first occurrence of an integer, n, in the parent sequence, is the first of a run of n^n elements of value n. For later occurrences, the run length is n^k-(n-1)^k where k is the size of the k-tuple in the grandparent sequence, A229873.

The elements can be arranged into a triangle thus:

.... 1

.... 1, 4

.... 1, 5, 27

.... 1, 7, 37, 256

.... 1, 9, 61, 369, 3125

.... etc.

where the n-th line is:

.... n^1-(n-1)^1, n^2-(n-1)^2, ..., n^(k-1)-(n-1)^(k-1), n^n; 1 <= k < n

The first terms, for sufficiently large n simplifying as:

.... 1, 2n-1, 3n^2-3n+1, 4n^3-6n^2+4n-1, etc.

Row sums are first differences of A031972, and thus the cumulative sum of rows at the end of each row is A031972 itself, i.e., n*(n^n - 1)/(n-1).