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Also semiperimeter of the n-th region of the geometric version of the section model of partitions. Note that a(n) is easily viewable as the sum of two perpendicular segments with a shared vertex. The horizontal segment has length A141285(n) and the vertical segment has length A194446(n). The difference between these two segments gives A194447(n). See also an illustration in the Links section. For the definition of "region" see A206437.

Also triangle read by rows: T(n,k) = largest part plus the number of parts of the k-th region of the last section of the set of partitions of n.

For the definition of "region" see A206437. See also A186114 and A193870. - Omar E. Pol, May 21 2021

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

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

_

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 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 = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y]. See below:

.

. j Diagram 1 Partitions Diagram 2

. _ _ _ _

. 15 | _ | 7 _ |

. 14 | _ | | 4+3 _ | |

. 13 | _ | | 5+2 _ | |

. 12 | _| |_ | 3+2+2 _| |_ |

. 11 | _ | | 6+1 _ | |

. 10 | _| | | 3+3+1 _ | | |

. 9 | | | | 4+2+1 | | |

. 8 | |_ | | | 2+2+2+1 |_ | | |

. 7 | _ | | | 5+1+1 _ | | |

. 6 | _| | | | 3+2+1+1 _ | | | |

. 5 | | | | | 4+1+1+1 | | | |

. 4 | |_ | | | | 2+2+1+1+1 |_ | | | |

. 3 | | | | | | 3+1+1+1+1 | | | | |

. 2 | | | | | | | 2+1+1+1+1+1 | | | | | |

. 1 |||_|||_|_| 1+1+1+1+1+1+1 | | | | | | |

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. 1 2 3 4 5 6 7

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The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.

Also the diagram has the property that it can be transformed in a Dyck path (see example).

The sequence gives the height of the infinite Dyck path after n-th step.

The absolute values of the first differences give A000012.

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

Q: Is this infinite Dyck path a fractal?