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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. 34
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

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)

CROSSREFS

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, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850.

Sequence in context: A091029 A272372 A280850 * A299778 A302248 A235773

Adjacent sequences:  A296505 A296506 A296507 * A296509 A296510 A296511

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Feb 10 2018

STATUS

approved

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Last modified May 8 19:28 EDT 2021. Contains 343666 sequences. (Running on oeis4.)