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Also largest part of the n-th region of the set of partitions of j, if 1 <= n <= A000041(j). For the definition of "region of the set of partitions of j" see A206437.

Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.

For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.

Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013

Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

For the definition of "region" of the set of partitions of j, see A206437.

a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.

Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.

a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012

From Omar E. Pol, Aug 19 2013: (Start)

In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).

a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.

(End)

The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.

a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016

a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.

To define the sequence we use the following rules:

We start in the first quadrant of the square grid with no toothpicks.

If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).

If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).

The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.

Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

Number of toothpicks added at n-th stage to the toothpick structure (related to integer compositions) of A228370.

The equivalent sequence for integer partitions is A220517.

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).

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:

.

j Diagram 1 Diagram 2

_

7 | _ | | _

6 | _| | | _ |

5 | | | | |

4 | |_ | | | |_ |

3 | | | | | | |

2 | | | | | | | | |

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

.

. 1 2 3 4 5

.

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 | | | | | | |

.

. 1 2 3 4 5 6 7

.

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?