A368399 Irregular triangle read by rows: row n lists the indices of rows of the Christmas tree pattern (A367508) of order n, sorted by row length and, in case of ties, by row index.

1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 5, 6, 1, 2, 4, 5, 7, 3, 6, 8, 9, 10, 1, 3, 7, 9, 13, 2, 4, 5, 8, 10, 11, 14, 15, 17, 6, 12, 16, 18, 19, 20, 1, 2, 4, 5, 7, 11, 12, 14, 15, 17, 21, 22, 24, 28, 3, 6, 8, 9, 13, 16, 18, 19, 23, 25, 26, 29, 30, 32, 10, 20, 27, 31, 33, 34, 35

Offset: 1

Author

Paolo Xausa, Dec 23 2023

Keywords

• nonn
• tabf

Comments

Row n is a permutation of the integers in the interval [1, binomial(n,floor(n/2))].

See A367508 for the description of the Christmas tree patterns, references and links.

Examples

			Triangle begins (vertical bars separate indices of rows having different lengths):
.
  [1]  1;
  [2]  1| 2;
  [3]  1  2| 3;
  [4]  1  3| 2  4  5| 6;
  [5]  1  2  4  5  7| 3  6  8  9|10;
  [6]  1  3  7  9 13| 2  4  5  8 10 11 14 15 17| 6 12 16 18 19|20;
  ...
For example, the order 4 of the Christmas tree pattern is the following:
.
            1010             Row 1 length = 1
       1000 1001 1011        Row 2 length = 3
            1100             Row 3 length = 1
       0100 0101 1101        Row 4 length = 3
       0010 0110 1110        Row 5 length = 3
  0000 0001 0011 0111 1111   Row 6 length = 5
.
and ordering the rows by length (and then by row index) gives 1, 3, 2, 4, 5, 6.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..13494 (rows 1..15 of the triangle, flattened).

Crossrefs

Cf. A001405, A363718 (row lengths), A367508, A368400.

Programs

Mathematica

With[{nmax=8},Map[Flatten[Values[PositionIndex[#]]]&,SubstitutionSystem[{1->{2},t_/;t>1:>{t-1,t+1}},{2},nmax-1]]]