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.

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

Offset: 1

Author

Omar E. Pol, Aug 29 2021

Keywords

• nonn
• tabf

Comments

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.

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.

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

In the ziggurat diagram of n we have that:

The number of parts equals A237271(n).

The number of subparts equals A001227(n).

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

The total number of steps equals A000203(n).

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).

Examples

			Triangle begins:
  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, 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, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
  ...
Written as an isosceles triangle we can see the symmetry of every row as shown below:
                                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, 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, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
     1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
...
For n = 15 the ziggurat diagram of 15 looks like this:
                                _
                               | |
                  _            | |            _
                _| |          _| |_          | |_
              _|   |         |     |         |   |_
            _|     |         |     |         |     |_
          _|       |        _|     |_        |       |_
        _|         |       |         |       |         |_
      _|           |       |         |       |           |_
    _|             |      _|    _    |_      |             |_
   |_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|
    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
.
Where B = 10 + 1 = 11.
The left-hand part (or the left-hand staircase) has 8 steps.
The right-hand part (or the right-hand staircase) has 8 steps.
The central part (formed by two subparts or two staircases) has a total of 7 + 1 = 8 steps.
The number of parts equals A237271(15) = 3.
The number of subparts equals A001227(15) = 4.
The number of steps in the central column equals A067742(15) = 2.
The total number of steps equals A000203(15) = 24.
Compare the above diagram with the symmetric representation of sigma(15) with subparts as shown below:
                                _
                               | |
                               | |
                               | |
                               | |
                               | |
                               | |
                               | |
                          _ _ _|_|
                      _ _| |      8
                     |  _ _|
                    _| |_|
                   |_ _|  1
                   |    7
    _ _ _ _ _ _ _ _|
   |_ _ _ _ _ _ _ _|
                    8
.
The left-hand part has 8 square cells.
The right-hand part has 8 square cells.
The central part (formed by two subparts) has a total of 7 + 1 = 8 square cells.
The number of parts equals A237271(15) = 3.
The number of subparts equals A001227(15) = 4.
The number of square cells on the main diagonal equals A067742(15) = 2.
The total number of square cells equals A000203(15) = 24.
		

Crossrefs

Row lengths give A005408.

Analog of A249351.

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