A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

1, 2, 1, 1, 3, 2, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 2, 1, 3, 5, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 3, 2, 2, 3, 1, 4, 6, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 5, 1, 2, 1, 2, 1, 2, 2, 4, 2, 1, 1, 3, 3, 3, 2, 4, 1, 5, 7, 2, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 2, 1

Offset: 0

Author

Gus Wiseman, Jul 14 2025

Keywords

• nonn
• tabf

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

This is the triangle A245563, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A245563.

Examples

			The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 2
   3: 1 1
   4: 3
   5: 2 1
   6: 1 2
   7: 4
   8: 1 1 1
   9: 3 1
  10: 2 2
  11: 1 3
  12: 5
  13: 2 1 1
  14: 1 2 1
  15: 4 1
  16: 1 1 2
  17: 3 2
  18: 2 3
  19: 1 4
  20: 6
  21: 1 1 1 1
		

Crossrefs

In the following references, "before" is short for "before removing duplicate rows".

Positions of singleton rows appear to be A000071 = A000045-1, before A023758.

Positions of firsts appearances appear to be A001629.

Positions of rows of the form (1,1,...) appear to be A055588 = A001906+1.

First term of each row appears to be A083368.

Row sums appear to be A200648, before A000120.

Row lengths after the first row appear to be A200650+1, before A069010 = A037800+1.

Before the removals we had A245563 (except first term), see A245562, A246029, A328592.

For anti-run ranks we have A385816, before A348366, firsts A052499.

Standard composition numbers of rows are A385818, before A385889.

For anti-runs we have A385886, before A384877, firsts A384878.

Cf. A044813, A048793, A164707, A200649, A242882, A243815, A384175, A384890.

Programs

Mathematica

DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}]]