A353850 Number of integer compositions of n with all distinct run-sums.

1, 1, 2, 4, 5, 12, 24, 38, 52, 111, 218, 286, 520, 792, 1358, 2628, 4155, 5508, 9246, 13182, 23480, 45150, 54540, 94986, 146016, 213725, 301104, 478586, 851506, 1302234, 1775482, 2696942, 3746894, 6077784, 8194466, 12638334, 21763463, 28423976, 45309850, 62955524, 94345474

Offset

0,3

Comments

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).

Links

Joseph Likar, Table of n, a(n) for n = 0..120

Example

The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (1111)  (41)
                                (113)
                                (122)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - Joseph Likar, Aug 04 2023

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]], {n, 0, 15}]

Crossrefs

For distinct parts instead of run-sums we have A032020.

For distinct multiplicities instead of run-sums we have A242882.

For distinct run-lengths instead of run-sums we have A329739, ptns A098859.

For runs instead of run-sums we have A351013.

For partitions we have A353837, ranked by A353838 (complement A353839).

For equal instead of distinct run-sums we have A353851, ptns A304442.

These compositions are ranked by A353852.

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

A003242 counts anti-run compositions, ranked by A333489.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A175413 lists numbers whose binary expansion has all distinct runs.

A351014 counts distinct runs in standard compositions, firsts A351015.

A353847 gives composition run-sum transformation.

A353929 counts distinct runs in binary expansion, firsts A353930.

Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

Sequence in context: A056984 A117556 A091071 * A050599 A281643 A102932

Adjacent sequences: A353847 A353848 A353849 * A353851 A353852 A353853

Keyword

nonn

Author

Gus Wiseman, May 31 2022

Extensions

Terms a(21) and onwards from Joseph Likar, Aug 04 2023

Status

approved