cp's OEIS Frontend

This is the "Mathematica" ordering of the partitions, referenced in numerous other sequences. The partitions of each integer are in reverse order of the conjugates of the partitions in Abramowitz and Stegun order (A036036). They are in the reverse of the order of the partitions in Maple order (A080576). - Franklin T. Adams-Watters, Oct 18 2006

The graded reverse lexicographic ordering of the partitions is often referred to as the "Canonical" ordering of the partitions. - Daniel Forgues, Jan 21 2011

Also the "MAGMA" ordering of the partitions. - Jason Kimberley, Oct 28 2011

Also an intuitive ordering described but not formalized in [Hardy and Wright] the first four editions of which precede [Abramowitz and Stegun]. - L. Edson Jeffery, Aug 03 2013

Also the "Sage" ordering of the partitions. - Peter Luschny, Aug 12 2013

While this is the order used for the constructive function "IntegerPartitions", it is different from Mathematica's canonical ordering of finite expressions, the latter giving A036036 if parts of partitions are read in reversed (weakly increasing) order, or A334301 if in the usual (weakly decreasing) order. - Gus Wiseman, May 08 2020

First differs from A334442 for reversed partitions of 9. Namely, this sequence has (1,4,4) before (2,2,5), while A334442 has (2,2,5) before (1,4,4). - Gus Wiseman, May 07 2020

This is the "Abramowitz and Stegun" ordering of the partitions, referenced in numerous other sequences. The partitions are in reverse order of the conjugates of the partitions in Mathematica order (A080577). Each partition is the conjugate of the corresponding partition in Maple order (A080576). - Franklin T. Adams-Watters, Oct 18 2006

The "Abramowitz and Stegun" ordering of the partitions is the graded reflected colexicographic ordering of the partitions. - Daniel Forgues, Jan 19 2011

The "Abramowitz and Stegun" ordering of partitions has been traced back to C. F. Hindenburg, 1779, in the Knuth reference, p. 38. See the Hindenburg link, pp. 77-5 with the listing of the partitions for n=10. This is also mentioned in the P. Luschny link. - Wolfdieter Lang, Apr 04 2011

The "Abramowitz and Stegun" order used here means that the partitions of a given number are listed by increasing number of (nonzero) parts, then by increasing lexicographical order with parts in (weakly) indecreasing order. This differs from n=9 on from A334442 which considers reverse lexicographic order of parts in (weakly) decreasing order. - M. F. Hasler, Jul 12 2015, corrected thanks to Gus Wiseman, May 14 2020

This is the Abramowitz-Stegun ordering of reversed partitions (finite weakly increasing sequences of positive integers). The same ordering of non-reversed partitions is A334301. - Gus Wiseman, May 07 2020

The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].

Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - Omar E. Pol, Aug 24 2013

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - Joerg Arndt, Sep 02 2013

First differs from A334439 for partitions of 9. Namely, this sequence has (4,4,1) before (5,2,2), while A334439 has (5,2,2) before (4,4,1). - Gus Wiseman, May 08 2020

This is also a list of all the possible prime signatures of a number, arranged in graded colexicographic ordering. - N. J. A. Sloane, Feb 09 2014

This is also the Abramowitz-Stegun ordering of reversed partitions (A036036) if the partitions are reversed again after sorting. Partitions sorted first by sum and then colexicographically are A211992. - Gus Wiseman, May 08 2020

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]

Written as a triangle; row n has length A006128(n); row sums give A066186. Also written as an irregular tetrahedron in which T(n,j,k) is the k-th largest part of the j-th partition of n; the sum of column k in the slice n is A181187(n,k); right border of the slices gives A182715. - Omar E. Pol, Mar 25 2012

The equivalent sequence for compositions (ordered partitions) is A228351. - Omar E. Pol, Sep 03 2013

This is the reverse-colexicographic order of integer partitions, or the reflected reverse-lexicographic order of reversed integer partitions. It is not reverse-lexicographic order (A080577), wherein we would have (3,1) before (2,2). - Gus Wiseman, May 12 2020

First differs from A036037 for partitions of 9. Namely, this sequence has (5,2,2) before (4,4,1), while A036037 has (4,4,1) before (5,2,2).

This is the Abramowitz-Stegun ordering of integer partitions (A334301) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334302.

First differs from A334438 (shifted left once) at a(75) = 98, A334438(76) = 99. - Gus Wiseman, May 20 2020

This mapping of the set of all partitions of N >= 0 to {1, 2, 3, ...} (set of natural numbers) is one to one (bijective). The empty partition for N = 0 maps to 1.

A129129 seems to be analogous, except that the partition ordering A080577 is used. This ordering, however, does not care about the number of parts: e.g., 1^2,4 = 4,1^2 comes before 3^2, so a(23)=28 and a(22)=25 are interchanged.

Also Heinz numbers of all reversed integer partitions (finite weakly increasing sequences of positive integers), sorted first by sum, then by length, and finally lexicographically, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The version for non-reversed partitions is A334433. - Gus Wiseman, May 20 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.

A permutation of the positive integers.

This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

The tree begins (at height n, n >= 0, nodes represent partitions of n)

0: 1

1: 2

2: 3 4

3: 5 6 8

4: 7 10 9 12 16

5: 11 14 15 20 18 24 32

...

and hence differs from A114622.

Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018

These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]

Also compositions in the triangle of A066099 that are in nondecreasing order.

The equivalent sequence for compositions (ordered partitions) is A066099.

Row n has length A006128(n).

Row sums give A066186.