cp's OEIS Frontend

Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.

Also the finite sequence a(1)..a(r), where a(r) is a record in the sequence, is also a finite triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of the integer whose number of partitions is equal to a(r).

T(n,k) is also 1 plus the number of holes between T(n,k) and the previous member in the column k of triangle.

T(n,k) is also the height of the column mentioned in the definition, in a three-dimensional model of the set of partitions of j, in which the regions appear rotated 90 degrees and where the pivots are the largest part of every region (see A141285). For the definition of "region" see A206437. - Omar E. Pol, Feb 06 2014

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?

Also 0 together the alternating sums of A220517.

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 height of the peaks and the valleys of the infinite Dyck path give this sequence.

Q: Is this Dyck path a fractal?

Also first differences of A211978.

The rows of the triangle in A244966 converge to this sequence.

This is the original sequence of a large number of sequences connected with the section model of partitions.

Here "the n-th section of the set of partitions of any integer greater than or equal to n" (hence "the last section of the set of partitions of n") is defined to be the set formed by all parts that occur as a result of taking all partitions of n and then removing all parts of the partitions of n-1. For integers greater than 1 the structure of a section has two main areas: the head and tail. The head is formed by the partitions of n that do not contain 1 as a part. The tail is formed by A000041(n-1) partitions of 1. The set of partitions of n contains the sets of partitions of the previous numbers. The section model of partitions has several versions according with the ordering of the partitions or with the representation of the sections. In this sequence we use the ordering of A026791.

The section model of partitions can be interpreted as a table of partitions. See also A138121. - Omar E. Pol, Nov 18 2009

It appears that the versions of the model show an overlapping of sections and subsections of the numbers congruent to k mod m into parts >= m. For example:

First generation (the main table):

Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.

Second generation:

Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.

Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.

Third generation:

Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.

Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.

Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.

And so on.

Conjecture:

Let j and n be integers congruent to k mod m such that 0 <= k < m <= j < n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the section model, h is the number of sections or subsections removed), (Omar E. Pol, Dec 05 2010, Dec 06 2010).

Starting from the first row of triangle, it appears that the total numbers of parts of size k in k successive rows give the sequence A000041 (see A182703). - Omar E. Pol, Feb 22 2012

The last section of n contains A187219(n) regions (see A206437). - Omar E. Pol, Nov 04 2012