A350357 - OEIS

A350357

Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

%I #51 Jan 18 2022 06:08:10

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

%T 1,3,1,2,1,1,6,3,3,4,2,2,2,2,1,1,1,1,1,1,1,1,2,3,6,1,2,4,1,3,1,2,1,2,

%U 1,1,7,4,3,5,2,3,2,2,1,1,1,1,1,1,1,1,1,1,1

%N Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

%e Triangle begins:

%e [1], [1];

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

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

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

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

%e ...

%e Illustration of the first six rows of triangle in an infinite table:

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

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

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

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

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

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

%e | P | | | | | | | 2 2 2 |

%e | A | | | | | | 5 | 1 |

%e | R | | | | | | 3 2 | 1 |

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

%e | S | | | | | 2 2 | 1 | 1 |

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

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

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

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

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

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

%e | V | A027750 | | | | 1 | 1 2 | 1 3 |

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

%e | S | A027750 | | | | | 1 | 1 2 |

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

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

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

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

%e .

%e For n = 6 in the upper zone of the above table we can see the parts of the last section of the set of partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A138121.

%e In the lower zone of the table we can see the terms from the 6th row of A336812, these are the divisors of the numbers from the 6th row of A336811.

%e Note that in the lower zone of the table every row gives A027750.

%e The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.

%e For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A336812 and A338156.

%e The growth of the upper zone of the table is in accordance with the growth of the modular prism described in A221529.

%e The growth of the lower zone of the table is in accordance with the growth of the tower described also in A221529.

%e The number of cubic cells added at n-th stage in each polycube is equal to A138879(10) = 150, hence the total number of cubic cells added at n-th stage is equal to 2*A138879(10) = 300, equaling the sum of the 10th row of this triangle.

%Y Companion of A350333.

%Y Row sums give 2*A138879.

%Y Row lengths give 2*A138137.

%Y Cf. A002865, A135010, A138121, A138879, A221529, A237593, A336812, A339278.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Dec 26 2021