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Let r = T(n,k) be a record in the sequence. The consecutive records "r" are the natural numbers A000027. Consider the first n rows; the triangle T(n,k) has the property that the columns, without the zeros, from k..1, are also the partitions of r in juxtaposed reverse-lexicographical order, so k is also A000041(r), the number of partitions of r. Note that a record r is always the final term of a row if such row contains 1’s. The number of positive integer a(1)..r is A006128(r). The sums a(1)..r is A066186(r). Here the set of positive integers in every row (from 1 to n) is called a “region” of r. The number of regions of r equals the number of partitions of r. If T(n,1) = 1 then the row n is formed by the smallest parts, in nondecreasing order, of all partitions of T(n,n).

Row sums of the triangles A135010, A138121, A138151 and others related to the section model of partitions (see A135010 and A138121).

From Omar E. Pol, Jan 20 2021: (Start)

Convolution of A000203 and A002865.

Convolution of A340793 and A000041.

Row sums of triangles A339278, A340426, A340583. (End)

a(n) is also the sum of all divisors of all terms of n-th row of A336811. These divisors are also all parts in the last section of the set of partitions of n. - Omar E. Pol, Jul 27 2021

Row sums of A336812. - Omar E. Pol, Aug 03 2021

Also total number of largest parts in all partitions of n. - Vladeta Jovovic, Feb 16 2004

To see this, consider the properties of a partition related through conjugation, such as the total number of parts and the size of the largest parts. The sums over all of the partitions of n of these two properties are equal. The size of the smallest part and the number of largest parts are two such properties (this is immediate when looking at the Ferrers diagram). - Michael Donatz, Apr 17 2011

Starting with offset 1, = the partition triangle A026794 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008

For n >= 1, a(n) = T(n+1,1) + T(n+2,2) + T(n+3,3)+ ... (sum along a falling diagonal) of the partition triangle A026794. - Bob Selcoe, Jun 22 2013

Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014

From Omar E. Pol, Jul 10 2021: (Start)

The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.

The base of the tower is the symmetric representation of A024916(n).

The height of the tower is equal to A000041(n-1).

The surface area of the tower is equal to A345023(n).

The volume (or the number of cubes) of the tower equals A066186(n).

The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).

Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].

The tower is an object of the family of the stepped pyramid described in A245092.

T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.

T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.

The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.

Therefore the set of all partitions of n >= 1 has an associated tower.

The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.

Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)

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

In other words: in row n replace every term of n-th row of A176206 with its divisors.

The terms in row n are also all parts of all partitions of n.

As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.

For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.

From Omar E. Pol, Aug 01 2021: (Start)

The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.

The terms of row n in nonincreasing order give the n-th row of A302246.

The terms of row n in nondecreasing order give the n-th row of A302247.

For the connection with the tower described in A221529 see also A340035. (End)

Number of different values of A007947(m) when A056239(m) is equal to n.

From Gus Wiseman, Sep 11 2023: (Start)

Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:

{1} {1} {1} {1} {1} {1} {1}

{2} {3} {2} {5} {2} {7}

{1,2} {4} {1,2} {3} {1,2}

{1,2} {1,3} {6} {1,3}

{1,3} {1,4} {1,2} {1,4}

{2,3} {1,3} {1,5}

{1,4} {1,6}

{1,5} {2,3}

{2,4} {2,5}

{1,2,3} {3,4}

{1,2,3}

{1,2,4}

Cf. A365073, A365311, A365312, A365322, A365380.

(End)

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.

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011

For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013