A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 0

Author

Gus Wiseman, May 31 2022

Keywords

• nonn

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Examples

			The terms together with their binary expansions and corresponding compositions begin:
   0:        0  ()
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   5:      101  (2,1)
   6:      110  (1,2)
   7:      111  (1,1,1)
   8:     1000  (4)
   9:     1001  (3,1)
  10:     1010  (2,2)
  12:     1100  (1,3)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  17:    10001  (4,1)
  18:    10010  (3,2)
  19:    10011  (3,1,1)
  20:    10100  (2,3)
  21:    10101  (2,2,1)
  23:    10111  (2,1,1,1)
		

Crossrefs

The version for runs in binary expansion is A175413.

The version for parts instead of run-sums is A233564, counted A032020.

The version for run-lengths instead of run-sums is A351596, counted A329739.

The version for runs instead of run-sums is A351290, counted by A351013.

The version for partitions is A353838, counted A353837, complement A353839.

The equal instead of distinct version is A353848, counted by A353851.

These compositions are counted by A353850.

The weak version (rucksack compositions) is A354581, counted by A354580.

A003242 counts anti-run compositions, ranked by A333489.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A242882 counts composition with distinct multiplicities, partitions A098859.

A304442 counts partitions with all equal run-sums.

A351014 counts distinct runs in standard compositions, firsts A351015.

A353853-A353859 pertain to composition run-sum trajectory.

A353864 counts rucksack partitions, perfect A353865.

A353929 counts distinct runs in binary expansion, firsts A353930.

Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Split[stc[#]]&]