A296508 - cp's OEIS frontend

A296508 Irregular triangle read by rows: T(n,k) is the size of the subpart that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned subpart is already associated to a previous peak or if there is no subpart adjacent to the k-th peak, with n >= 1, k >= 1.

1, 3, 2, 2, 7, 0, 3, 3, 11, 1, 0, 4, 0, 4, 15, 0, 0, 5, 3, 5, 9, 0, 9, 0, 6, 0, 0, 6, 23, 5, 0, 0, 7, 0, 0, 7, 12, 0, 12, 0, 8, 7, 1, 0, 8, 31, 0, 0, 0, 0, 9, 0, 0, 0, 9, 35, 2, 0, 2, 0, 10, 0, 0, 0, 10, 39, 0, 3, 0, 0, 11, 5, 0, 5, 0, 11, 18, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 12, 47, 13, 0, 0, 0, 0, 13, 0, 5, 0, 0, 13

Offset: 1

Omar E. Pol, Feb 10 2018

Conjecture: row n is formed by the odd-indexed terms of the n-th row of triangle A280850 together with the even-indexed terms of the same row but listed in reverse order. Examples: the 15th row of A280850 is [8, 8, 7, 0, 1] so the 15th row of this triangle is [8, 7, 1, 0, 8]. The 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0] so the 75h row of this triangle is [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38].
For the definition of "subparts" see A279387.
For more information about the mentioned Dyck paths see A237593.
T(n,k) could be called the "charge" of the k-th peak of the largest Dyck path of the symmetric representation of sigma(n).
The number of zeros in row n is A238005(n). - Omar E. Pol, Sep 11 2021

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  11,  1,  0;
   4,  0,  4;
  15,  0,  0;
   5,  3,  5;
   9,  0,  9,  0;
   6,  0,  0,  6;
  23,  5,  0,  0;
   7,  0,  0,  7;
  12,  0, 12,  0;
   8,  7,  1,  0,  8;
  31,  0,  0,  0,  0;
   9,  0,  0,  0,  9;
  35,  2,  0,  2,  0;
  10,  0,  0,  0, 10;
  39,  0,  3,  0,  0;
  11,  5,  0,  5,  0, 11;
  18,  0,  0,  0, 18,  0;
  12,  0,  0,  0,  0, 12;
  47, 13,  0,  0,  0,  0;
  13,  0,  5,  0,  0, 13;
  21,  0,  0,  0  21,  0;
  14,  6,  0,  6,  0, 14;
  55,  0,  0,  1,  0,  0,  0;
  ...
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) is constructed in the third quadrant as shown below in Figure 1:
.    _                                  _
.   | |                                | |
.   | |                                | |
.   | |                                | |
. 8 | |                                | |
.   | |                                | |
.   | |                                | |
.   | |                                | |
.   |_|_ _ _                           |_|_ _ _
.         | |_ _                      8      | |_ _
.         |_    |                            |_ _  |
.           |_  |_                          7  |_| |_
.          8  |_ _|                           1  |_ _|
.                 |                             0    |
.                 |_ _ _ _ _ _ _ _                   |_ _ _ _ _ _ _ _
.                 |_ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _|
.                         8                         8
.
.   Figure 1. The symmetric            Figure 2. After the dissection
.   representation of sigma(15)        of the symmetric representation
.   has three parts of size 8          of sigma(15) into layers of
.   because every part contains        width 1 we can see four subparts,
.   8 cells, so the 15th row of        so the 15th row of this triangle is
.   triangle A237270 is [8, 8, 8].     [8, 7, 1, 0, 8]. See also below.
.
Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.              0 _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _ 0
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
.    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |    0 _ _| |   11 _ _ _ _          |_  |         | |
.     | |     |  _ _|  1 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |    0 _|_| |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|0    | |     | |
.   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|  _ _ _|0    | |
.   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|  _| |    _ _ _| |
.  8      | |_ _   0  |                     15|  _|  _|   |  _ _ _|
.         |_ _  |     |_ _ _ _ _ _            | |_ _|  0 _| |      0
.        7  |_| |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    5 _|  _|
.          1  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
.            0    |                             23|  _ _|  0
.                 |_ _ _ _ _ _ _ _                | |    0
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains 30 subparts equaling A060831(16), the total number of partitions of all positive integers <= 16 into consecutive parts.
For the construction of the spiral see A239660.
From _Omar E. Pol_, Nov 26 2020: (Start)
Also consider the infinite double-staircases diagram defined in A335616 (see the theorem). For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
Starting from A196020 and after the algorithm described n A280850 and the conjecture applied to the above diagram we have a new diagram as shown below:
.
Level                             "Ziggurat" diagram
.                                          _
6                                         |1|
7                            _            | |            _
8                          _|1|          _| |_          |1|_
9                        _|1  |         |1   1|         |  1|_
10                     _|1    |         |     |         |    1|_
11                   _|1      |        _|     |_        |      1|_
12                 _|1        |       |1       1|       |        1|_
13               _|1          |       |         |       |          1|_
14             _|1            |      _|    _    |_      |            1|_
15            |1              |     |1    |1|    1|     |              1|
.
The 15th row
of A249351:   [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
The 15th row
of A237270:   [              8,            8,            8              ]
The 15th row
of this seq:  [              8,      7,    1,    0,      8              ]
The 15th row
of A280851:   [              8,      7,    1,            8              ]
.
(End)

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
Nonzero terms give A280851.
The number of nonzero terms in row n is A001227(n).
The triangle with n rows contain A060831(n) nonzero terms.
Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A238005, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850.

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