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Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(k) and the k-th vertical line segment has length A194446(k). Both line segments shares the point (A141285(k),k). For n>=1, the table contains A000041(n) regions which are distributed in n sections. Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first n sections from the table we have a representation of the partitions of n. For an illustration see the example. For the definition of "region" see A206437. For the definition of "section" see A135010. For a visualization of the corner of size n X n of the table see A273140.

a(n) is also the sum of the perimeters of the Ferrers boards of the partitions of n, minus the sum of the perimeters of the Ferrers boards of the partitions of n-1, with n >= 1. For more information see A278355.

The second largest part of the n-th partition of the table A182742.

In n is odd then row n lists the numbers of row n of A135010. If n is even the row n lists the 1's of row n of A135010 and then row n lists the other numbers of row n of A135010 in reverse order.

For more information see the example in A135010.

Volumes of the parts in the section model of partitions version "boxes" in which each part of size k has a volume = k^2. Row sums of this triangle give A206440 and partial sums of A206440 give A066183.

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

On the infinite square grid the diagram of regions of the set of partitions of n is represented by a rectangle with base = n and height = A000041(n). The rectangle contains n shells. Each shell contains regions. Each row of a region is a part. Each part of size k contains k cells. The number of regions equals the number of partitions of n (see illustrations in the links section). For a minimalist version see A139582. For the definition of "region of n" see A206437.

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.

For k = 5, the diagram 1 represents the partitions of 5. The diagram 2 shows separately the boundary segments southwest sides of the first seven regions of the diagram 1, see below:

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j Diagram 1 Diagram 2

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7 | _ | | _

6 | _| | | _ |

5 | | | | |

4 | |_ | | | |_ |

3 | | | | | | |

2 | | | | | | | | |

1 |||_||| | | | | |_

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. 1 2 3 4 5

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a(n) is the height after n-th step of an infinite staircase which is the lower part of a diagram of regions of the set of partitions of all positive integers. The upper part of the diagram is the infinite Dyck path mentioned in A228110. The diagram shows the shape of the successive regions of the set of partitions of all positive integers. The area of the n-th region is A186412(n).

For the height of the peaks and the valleys in the infinite Dyck path see A229946.

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

It appears that if this is written as a triangle (see example) and n >= 3 then row n has the following property:

If n is congruent to 0 (mod 3) then row n converge to the sequence 3,6,5,9,4,8,7,6,12... in which the records are the numbers >= 3 that are congruent to 0 (mod 3).

If n is congruent to 1 (mod 3) then row n converge to the sequence 4,7,6,5,10,5,9,8,7,13... in which the records are the numbers >= 4 that are congruent to 1 (mod 3).

If n is congruent to 2 (mod 3) then row n converge to the sequence 5,4,8,7,6,11,6,5,10,9,8,7,14... in which the records are the numbers >= 5 that are congruent to 2 (mod 3).

For more information see A135010.