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A347367 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.

%I A347367 #92 Sep 06 2023 11:49:22
%S A347367 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 A347367 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 A347367 5,0,0,1,4,1,0,0,5,4,3,2,1,1,2,3,4,5,6,7,8,9,0
%N A347367 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 A347367 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 A347367 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 A347367 The ziggurat diagram of n gives us an explanation about the parts, subparts and widths of the symmetric representation of sigma(n).
%C A347367 In the ziggurat diagram of n we have that:
%C A347367 The number of parts equals A237271(n).
%C A347367 The number of subparts equals A001227(n).
%C A347367 The number of steps in the central column equals A067742(n).
%C A347367 The total number of steps equals A000203(n).
%C A347367 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 A347367 Triangle begins:
%e A347367   1;
%e A347367   1, 2, 1;
%e A347367   1, 2, 0, 2, 1;
%e A347367   1, 2, 3, 4, 3, 2, 1;
%e A347367   1, 2, 3, 0, 0, 0, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
%e A347367   ...
%e A347367 Written as an isosceles triangle we can see the symmetry of every row as shown below:
%e A347367                                 1;
%e A347367                              1, 2, 1;
%e A347367                           1, 2, 0, 2, 1;
%e A347367                        1, 2, 3, 4, 3, 2, 1;
%e A347367                     1, 2, 3, 0, 0, 0, 3, 2, 1;
%e A347367                  1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;
%e A347367               1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;
%e A347367            1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;
%e A347367         1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
%e A347367      1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
%e A347367   1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
%e A347367 ...
%e A347367 For n = 15 the ziggurat diagram of 15 looks like this:
%e A347367                                 _
%e A347367                                | |
%e A347367                   _            | |            _
%e A347367                 _| |          _| |_          | |_
%e A347367               _|   |         |     |         |   |_
%e A347367             _|     |         |     |         |     |_
%e A347367           _|       |        _|     |_        |       |_
%e A347367         _|         |       |         |       |         |_
%e A347367       _|           |       |         |       |           |_
%e A347367     _|             |      _|    _    |_      |             |_
%e A347367    |_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|
%e A347367     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 A347367 .
%e A347367 Where B = 10 + 1 = 11.
%e A347367 The left-hand part (or the left-hand staircase) has 8 steps.
%e A347367 The right-hand part (or the right-hand staircase) has 8 steps.
%e A347367 The central part (formed by two subparts or two staircases) has a total of 7 + 1 = 8 steps.
%e A347367 The number of parts equals A237271(15) = 3.
%e A347367 The number of subparts equals A001227(15) = 4.
%e A347367 The number of steps in the central column equals A067742(15) = 2.
%e A347367 The total number of steps equals A000203(15) = 24.
%e A347367 Compare the above diagram with the symmetric representation of sigma(15) with subparts as shown below:
%e A347367                                 _
%e A347367                                | |
%e A347367                                | |
%e A347367                                | |
%e A347367                                | |
%e A347367                                | |
%e A347367                                | |
%e A347367                                | |
%e A347367                           _ _ _|_|
%e A347367                       _ _| |      8
%e A347367                      |  _ _|
%e A347367                     _| |_|
%e A347367                    |_ _|  1
%e A347367                    |    7
%e A347367     _ _ _ _ _ _ _ _|
%e A347367    |_ _ _ _ _ _ _ _|
%e A347367                     8
%e A347367 .
%e A347367 The left-hand part has 8 square cells.
%e A347367 The right-hand part has 8 square cells.
%e A347367 The central part (formed by two subparts) has a total of 7 + 1 = 8 square cells.
%e A347367 The number of parts equals A237271(15) = 3.
%e A347367 The number of subparts equals A001227(15) = 4.
%e A347367 The number of square cells on the main diagonal equals A067742(15) = 2.
%e A347367 The total number of square cells equals A000203(15) = 24.
%Y A347367 Row lengths give A005408.
%Y A347367 Analog of A249351.
%Y A347367 Cf. A000203, A001227, A067742, A196020, A235791, A236104, A237270, A237271, A237591, A237593 (Dyck paths), A245092, A279387 (subparts), A280850 (algorithm), A280851, A296508, A335616.
%K A347367 nonn,tabf
%O A347367 1,3
%A A347367 _Omar E. Pol_, Aug 29 2021