cp's OEIS Frontend

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.

%I A368399 #19 Jun 12 2024 03:37:50
%S A368399 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,
%T A368399 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,
%U A368399 6,8,9,13,16,18,19,23,25,26,29,30,32,10,20,27,31,33,34,35
%N 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.
%C A368399 Row n is a permutation of the integers in the interval [1, binomial(n,floor(n/2))].
%C A368399 See A367508 for the description of the Christmas tree patterns, references and links.
%H A368399 Paolo Xausa, <a href="/A368399/b368399.txt">Table of n, a(n) for n = 1..13494</a> (rows 1..15 of the triangle, flattened).
%e A368399 Triangle begins (vertical bars separate indices of rows having different lengths):
%e A368399 .
%e A368399   [1]  1;
%e A368399   [2]  1| 2;
%e A368399   [3]  1  2| 3;
%e A368399   [4]  1  3| 2  4  5| 6;
%e A368399   [5]  1  2  4  5  7| 3  6  8  9|10;
%e A368399   [6]  1  3  7  9 13| 2  4  5  8 10 11 14 15 17| 6 12 16 18 19|20;
%e A368399   ...
%e A368399 For example, the order 4 of the Christmas tree pattern is the following:
%e A368399 .
%e A368399             1010             Row 1 length = 1
%e A368399        1000 1001 1011        Row 2 length = 3
%e A368399             1100             Row 3 length = 1
%e A368399        0100 0101 1101        Row 4 length = 3
%e A368399        0010 0110 1110        Row 5 length = 3
%e A368399   0000 0001 0011 0111 1111   Row 6 length = 5
%e A368399 .
%e A368399 and ordering the rows by length (and then by row index) gives 1, 3, 2, 4, 5, 6.
%t A368399 With[{nmax=8},Map[Flatten[Values[PositionIndex[#]]]&,SubstitutionSystem[{1->{2},t_/;t>1:>{t-1,t+1}},{2},nmax-1]]]
%Y A368399 Cf. A001405, A363718 (row lengths), A367508, A368400.
%K A368399 nonn,tabf
%O A368399 1,3
%A A368399 _Paolo Xausa_, Dec 23 2023