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Irregular triangle read by rows: T(n,k) is the total number of cells with multiplicity in the k-th column of the ziggurat diagram of n.
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%I #92 Sep 06 2023 11:49:22

%S 1,1,2,1,1,2,0,2,1,1,2,3,4,3,2,1,1,2,3,0,0,0,3,2,1,1,2,3,4,5,7,5,4,3,

%T 2,1,1,2,3,4,0,0,0,0,0,4,3,2,1,1,2,3,4,5,6,7,8,7,6,5,4,3,2,1,1,2,3,4,

%U 5,0,0,1,4,1,0,0,5,4,3,2,1,1,2,3,4,5,6,7,8,9,0

%N Irregular triangle read by rows: T(n,k) is the total number of cells with multiplicity in the k-th column of the ziggurat diagram of n.

%C The "ziggurat" diagram arises as a remnant of the double-staircases diagram described in A335616 after a geometric algorithm equivalent to the algorithm described in A280850 and A296508.

%C The geometric algorithm is also equivalent to the folding of the isosceles triangle described in A237593 forming the structure of the pyramid described in A245092.

%C The ziggurat diagram of n gives us an explanation about the parts, subparts and widths of the symmetric representation of sigma(n).

%C In the ziggurat diagram of n we have that:

%C The number of parts equals A237271(n).

%C The number of subparts equals A001227(n).

%C The number of steps in the central column equals A067742(n).

%C The total number of steps equals A000203(n).

%C The correspondence between both diagrams is because a three-dimensional version of the ziggurat of n can be constructed with units cubes and where the base of the structure is the symmetric representation of sigma(n).

%e Triangle begins:

%e 1;

%e 1, 2, 1;

%e 1, 2, 0, 2, 1;

%e 1, 2, 3, 4, 3, 2, 1;

%e 1, 2, 3, 0, 0, 0, 3, 2, 1;

%e 1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;

%e ...

%e Written as an isosceles triangle we can see the symmetry of every row as shown below:

%e 1;

%e 1, 2, 1;

%e 1, 2, 0, 2, 1;

%e 1, 2, 3, 4, 3, 2, 1;

%e 1, 2, 3, 0, 0, 0, 3, 2, 1;

%e 1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;

%e 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;

%e ...

%e For n = 15 the ziggurat diagram of 15 looks like this:

%e _

%e | |

%e _ | | _

%e _| | _| |_ | |_

%e _| | | | | |_

%e _| | | | | |_

%e _| | _| |_ | |_

%e _| | | | | |_

%e _| | | | | |_

%e _| | _| _ |_ | |_

%e |_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|

%e 1 2 3 4 5 6 7 8 0 0 0 1 4 7 B 7 4 1 0 0 0 8 7 6 5 4 3 2 1

%e .

%e Where B = 10 + 1 = 11.

%e The left-hand part (or the left-hand staircase) has 8 steps.

%e The right-hand part (or the right-hand staircase) has 8 steps.

%e The central part (formed by two subparts or two staircases) has a total of 7 + 1 = 8 steps.

%e The number of parts equals A237271(15) = 3.

%e The number of subparts equals A001227(15) = 4.

%e The number of steps in the central column equals A067742(15) = 2.

%e The total number of steps equals A000203(15) = 24.

%e Compare the above diagram with the symmetric representation of sigma(15) with subparts as shown below:

%e _

%e | |

%e | |

%e | |

%e | |

%e | |

%e | |

%e | |

%e _ _ _|_|

%e _ _| | 8

%e | _ _|

%e _| |_|

%e |_ _| 1

%e | 7

%e _ _ _ _ _ _ _ _|

%e |_ _ _ _ _ _ _ _|

%e 8

%e .

%e The left-hand part has 8 square cells.

%e The right-hand part has 8 square cells.

%e The central part (formed by two subparts) has a total of 7 + 1 = 8 square cells.

%e The number of parts equals A237271(15) = 3.

%e The number of subparts equals A001227(15) = 4.

%e The number of square cells on the main diagonal equals A067742(15) = 2.

%e The total number of square cells equals A000203(15) = 24.

%Y Row lengths give A005408.

%Y Analog of A249351.

%Y Cf. A000203, A001227, A067742, A196020, A235791, A236104, A237270, A237271, A237591, A237593 (Dyck paths), A245092, A279387 (subparts), A280850 (algorithm), A280851, A296508, A335616.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Aug 29 2021