%I #91 Feb 16 2023 16:26:32

%S 1,1,2,1,3,1,1,2,4,1,2,1,1,5,1,3,1,2,1,1,1,2,3,6,1,2,4,1,3,1,2,1,2,1,

%T 1,1,7,1,5,1,2,4,1,3,1,3,1,2,1,2,1,1,1,1,1,2,4,8,1,2,3,6,1,5,1,2,4,1,

%U 2,4,1,3,1,3,1,2,1,2,1,2,1,2,1,1,1,1,1,3,9,1,7,1,2,3,6

%N Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

%C Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section).

%C For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.

%H Paolo Xausa, <a href="/A336812/b336812.txt">Table of n, a(n) for n = 1..10206 (rows 1..23 of triangle, flattened)</a>

%e Triangle begins:

%e [1];

%e [1, 2];

%e [1, 3], [1];

%e [1, 2, 4], [1, 2], [1];

%e [1, 5], [1, 3], [1, 2], [1], [1];

%e [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1];

%e ...

%e For n = 6 the 6th row of A336811 is [6, 4, 3, 2, 2, 1, 1] so replacing every term with its divisors we have {[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]} the same as the 6th row of this triangle.

%e Also, if the sequence is written as an irregular tetrahedron so the first six slices are:

%e -------------

%e [1],

%e -------------

%e [1, 2];

%e -------------

%e [1, 3],

%e [1];

%e -------------

%e [1, 2, 4],

%e [1, 2],

%e [1];

%e -------------

%e [1, 5],

%e [1, 3],

%e [1, 2],

%e [1],

%e [1];

%e -------------

%e [1, 2, 3, 6],

%e [1, 2, 4],

%e [1, 3],

%e [1, 2],

%e [1, 2],

%e [1],

%e [1];

%e -------------

%e The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and the parts of the last section of the set of partitions of the positive integers.

%e The table is infinite. It is formed by three zones as follows:

%e The upper zone shows the last section of the set of partitions of every positive integer.

%e The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.

%e Finally the middle zone shows the connection between the upper zone and the lower zone.

%e For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e | n | | 1 | 2 | 3 | 4 | 5 | 6 |

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e | | | | | | | | 6 |

%e | P | | | | | | | 3 3 |

%e | A | | | | | | | 4 2 |

%e | R | | | | | | | 2 2 2 |

%e | T | | | | | | 5 | 1 |

%e | I | | | | | | 3 2 | 1 |

%e | T | | | | | 4 | 1 | 1 |

%e | I | | | | | 2 2 | 1 | 1 |

%e | O | | | | 3 | 1 | 1 | 1 |

%e | N | | | 2 | 1 | 1 | 1 | 1 |

%e | S | | 1 | 1 | 1 | 1 | 1 | 1 |

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e .

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e | | A207031 | 1 | 2 1 | 3 1 1 | 6 3 1 1 | 8 3 2 1 1 | 15 8 4 2 1 1 |

%e | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| | |/|/|/|/|/| |

%e | I | A182703 | 1 | 1 1 | 2 0 1 | 3 2 0 1 | 5 1 1 0 1 | 7 4 2 1 0 1 |

%e | N | | * | * * | * * * | * * * * | * * * * * | * * * * * * |

%e | K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 | 1 2 3 4 5 6 |

%e | | | = | = = | = = = | = = = = | = = = = = | = = = = = = |

%e | | A207383 | 1 | 1 2 | 2 0 3 | 3 4 0 4 | 5 2 3 0 5 | 7 8 6 4 0 6 |

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e .

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 | 1 2 3 6 |

%e | D |---------|-----|-------|---------|-----------|-------------|---------------|

%e | I | A027750 | | | 1 | 1 2 | 1 3 | 1 2 4 |

%e | V |---------|-----|-------|---------|-----------|-------------|---------------|

%e | I | A027750 | | | | 1 | 1 2 | 1 3 |

%e | S |---------|-----|-------|---------|-----------|-------------|---------------|

%e | O | A027750 | | | | | 1 | 1 2 |

%e | R | A027750 | | | | | 1 | 1 2 |

%e | S |---------|-----|-------|---------|-----------|-------------|---------------|

%e | | A027750 | | | | | | 1 |

%e | | A027750 | | | | | | 1 |

%e |---|---------|-----|-------|---------|-----------|-------------|---------------|

%e .

%e Note that every row in the lower zone lists A027750.

%e The "section" is the simpler substructure of the set of partitions of n that has this property in the three zones.

%e Also the lower zone for every positive integer can be constructued using the first n terms of A002865. For example: for n = 6 we consider the first 6 terms of A002865 (that is [1, 0, 1, 1, 2, 2] and then the 6th slice is formed by a block with the divisors of 6, no block with the divisors of 5, one block with the divisors of 4, one block with the divisors of 3, two block with the divisors of 2 and two blocks with the divisors of 1.

%e Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.

%e The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).

%e The above table shows the growth step by step of both the prism of partitions and its associated tower since the number of parts in the last section of the set of partitions of n is equal to A138137(n) equaling the number of divisors in the n-th slice of the lower table and equalimg the same the number of terms in the n-th row of triangle. Also the sum of all parts in the last section of the set of partitions of n is equal to A138879(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

%t A336812[row_]:=Flatten[Table[ConstantArray[Divisors[row-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];

%t Array[A336812,10] (* Generates 10 rows *) (* _Paolo Xausa_, Feb 16 2023 *)

%Y Companion and subsequence of A338156.

%Y Row lengths give A138137.

%Y Row sums give A138879.

%Y Cf. A000041, A000070, A002260, A002865, A006128, A024916, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A187219, A207031, A207038, A207383, A221529, A221530, A221531, A237593, A245095, A221649, A221650, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A350357.

%K nonn,look,tabf

%O 1,3

%A _Omar E. Pol_, Nov 20 2020