cp's OEIS Frontend

First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.

A permutation of the positive integers.

This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435.

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.

a(0) = 0 by convention. - Franklin T. Adams-Watters, Jun 24 2014

Like A036043 this is important for calculating sequences defined over the numeric partitions, cf. A000041. For example, the triangular array A019575 can be calculated using A036042 and this sequence.

The row sums are A006128. - Johannes W. Meijer, Jun 21 2010

The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A334441. - Gus Wiseman, May 21 2020

T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).

The row length sequence of this table is p(n)=A000041(n) (number of partitions).

In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.

Rows are the same as the rows of A115623, but in reverse order.

From Wolfdieter Lang, Mar 17 2011: (Start)

The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.

For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).

See a comment by Giovanni Resta under A000005. (End)

From Gus Wiseman, May 20 2020: (Start)

The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.

Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).

Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.

(End)

A permutation of the positive integers.

This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).

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},{4,3},{8,6,5},...}, so offset is 0.

First differs from A334435 at a(22) = 27, A334435(22) = 22.

A permutation of the positive integers.

Reversed integer partitions are finite weakly increasing sequences of positive integers. For non-reversed partitions, see A129129 and A228531.

This is the so-called "Mathematica" order (A080577).

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.

A permutation of the positive integers.

Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.

This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).

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

Also Heinz numbers of partitions in colexicographic order (cf. A211992).

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

First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).

First differs from A049085 at a(8) = 2, A049085(8) = 3.

The parts of a partition are read in the usual (weakly decreasing) order. The version for reversed (weakly increasing) partitions is A049085.

The row lengths sequence is A000041(n), n>=1.

The partitions are ordered like in Abramowitz-Stegun (A-St order). For the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used.

A necklace with n beads (n-necklace) has here only the cyclic C_n symmetry group. This is in contrast to, e.g., the Harary-Palmer reference, p. 44, where a n-necklace has the symmetry group D_n, the dihedral group of degree n (order 2n), which allows, in addition to C_n operations, also a turnover (in 3-space) or a reflection (in 2-space).

The necklace number a(n,k) gives the number of n-necklaces, with up to n colors for each bead, belonging to the k-th partition of n in A-St order in the following way. Write this partition with nonincreasing parts (this is the reverse of the partition as given by A-St), e.g., [3,1^2], not [1^2,3], is written as [3,1,1], a partition of n=5. In general [p[1],p[2],...,p[m]], with p[1]>=p[2]>=...>=p[m]>=1, with m the number of parts. To each such partition of n corresponds an n-multiset obtained by 'exponentiation'. For the given example the 5-multiset is {1^3,2^1,3^1}={1,1,1,2,3}. In general {1^p[1],2^p[2],...,m^p[m]}. Such an n-multiset representative (of a repetition class defined by the exponents, sometimes called signature) encodes the n-necklace color monomial by c[1]^p[1]*c[2]^p[2]*...*c[m]^p[m]. For the example one has c[1]^3*c[2]*c[3]. The number of 5-necklaces with this color assignment is a(5,4) because [3,1,1] is the 4th partition of 5 in A-St order. The a(5,4)=4 non-equivalent 5-necklaces with this color assignment are cyclic(c[1]c[1]c[1]c[2]c[3]), cyclic(c[1]c[1]c[1]c[3]c[2]), cyclic(c[1]c[1]c[2]c[1]c[3]) and cyclic(c[1]c[1]c[3]c[1]c[2]).

Such a set of a(n,k) n-necklaces for the given color assignment stands for other sets of the same order where different colors from the repertoire {c[1],...,c[n]} are chosen. In the example, the partition [3,1,1] with the representative multiset [1^3,2,3] stands for all-together 5*binomial(4,2) = 30 such sets, each leading to 4 possible non-equivalent 5-necklace arrangements. Thus one has, in total, 30*4=120 5-necklaces with color signature determined from the partition [3,1,1]. See the partition array A212360 for these numbers.

For the example n=4, k=1..5, see the Stanley reference, last line, where the numbers a(4,k) are, in A-St order, 1, 1, 2, 3, 6, summing to A072605(4).

a(n,k) is computed from the cycle index Z(C_n) for the cyclic group (see A212357 and the link given there) after the variables x_j have been replaced by the j-th power sum sum(c[i]^j,i=1..n), abbreviated as Z(C_n,c_n) with c_n:=sum(c[i],i=1..n), n>=1. The coefficient of the color assignment representative determined by the k-th partition of n in A-St order, as explained above, is a(n,k). See the Harary-Palmer reference, p. 36, Theorem (PET) with A=C_n and p. 36 eq. (2.2.10) for the cycle index polynomial Z(C_n). See the W. Lang link for more details.

The corresponding triangle with summed entries of row n which belong to partitions of n with the same number of parts is A213934. [Wolfdieter Lang, Jul 12 2012]

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.