A353847 -id:A353847 - cp's OEIS frontend

A354582 Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 2, 3, 5, 2, 2, 3, 3, 3, 3, 2, 4, 3, 3, 4, 3, 4, 4, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 2, 3, 4, 5, 2, 3, 2, 4, 3, 4, 3

Offset: 0

Gus Wiseman, Jun 13 2022

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.

			Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), so a(981) = 6.
As a triangle:
  1
  1 2
  1 2 2 3
  1 2 2 3 2 2 3 4
  1 2 2 3 2 3 2 4 2 2 3 3 3 3 4 5
  1 2 2 3 2 3 3 4 2 3 3 4 3 2 3 5 2 2 3 3 3 3 2 4 3 3 4 3 4 4 5 6

The version for partitions is A001222, full A001221.
If we allow any constant subsequence we get A063787.
If we allow any contiguous subsequence we get A124771.
Positions of first appearances are A126646.
The version for binary indices is A330036, full A005811.
If we allow any subsequence we get A334299.
The full version is A351014, firsts A351015.
The version for run-sums of partitions is A353861, full A353835.
Counting distinct sums of partial runs gives A354907, full A353849.
A066099 lists all compositions in standard order.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A000120, A003242, A029837, A175413, A274174, A333381, A333489, A353832, A353860, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
Table[Length[Union[Join@@pre/@Split[stc[n]]]],{n,0,100}]

A354909 Number of integer compositions of n that are not the run-sums of any other composition.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 16, 33, 74, 155, 329, 688, 1439, 2975, 6154, 12654, 25964, 53091, 108369, 220643, 448520

Offset: 0

Gus Wiseman, Jun 19 2022

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).

			The a(0) = 0 through a(6) = 16 compositions:
  .  .  (11)  (111)  (112)   (113)    (114)
                     (211)   (311)    (411)
                     (1111)  (1112)   (1113)
                             (1121)   (1122)
                             (1211)   (1131)
                             (2111)   (1221)
                             (11111)  (1311)
                                      (2112)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

The version for binary words is A000918, complement A000126.
These compositions are ranked by A354904 = positions of zeros in A354578.
The complement is counted by A354910, ranked by A354912.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A124767, A274174, A351597, A353849, A353850, A353860, A354905, A354907.

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

A354910 Number of compositions of n that are the run-sums of some other composition.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 16, 31, 54, 101, 183, 336, 609, 1121, 2038, 3730, 6804, 12445, 22703, 41501, 75768

Offset: 0

Gus Wiseman, Jun 20 2022

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).

			The a(0) = 0 through a(6) = 16 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (22)   (23)   (24)
                      (31)   (32)   (33)
                      (121)  (41)   (42)
                             (122)  (51)
                             (131)  (123)
                             (212)  (132)
                             (221)  (141)
                                    (213)
                                    (222)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (2121)

The version for binary words is A000126, complement A000918
The complement is counted by A354909, ranked by A354904.
These compositions are ranked by A354912 = nonzeros of A354578.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A239312, A274174, A351014, A351597, A353849, A353850, A353864, A354905, A354907.

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

A357181 Last run-length of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 3.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For parts instead of run-lengths we have A001511, first A065120.
For Heinz numbers of partitions we have A071178, first A067029.
This is the last part of row n of A333769.
For maximal instead of last we have A357137, minimal A357138.
The first instead of last run-length is A357180.
A051903 gives maximal part of prime signature.
A061395 gives maximal prime index.
A124767 counts runs in standard compositions.
A286470 gives maximal difference of prime indices.
A333766 gives maximal part of standard composition, minimal A333768.
A353847 ranks run-sums of standard compositions.
Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Last[Length/@Split[stc[n]]]],{n,0,100}]

A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 43, 46, 47, 58, 59, 60, 62, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 171, 174, 175, 184, 186, 187, 190, 191, 232, 234, 235, 238, 239, 248, 250, 251, 254, 255, 256, 292, 295, 316, 319, 484

Offset: 1

Gus Wiseman, Jun 23 2022

nonn

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.

If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.

			The terms together with their corresponding compositions begin:
  1:(1)  2:(2)   4:(3)     8:(4)     16:(5)      32:(6)
         3:(11)  7:(111)  10:(22)    31:(11111)  36:(33)
                          11:(211)               39:(3111)
                          14:(112)               42:(222)
                          15:(1111)              43:(2211)
                                                 46:(2112)
                                                 47:(21111)
                                                 58:(1122)
                                                 59:(11211)
                                                 60:(1113)
                                                 62:(11112)
                                                 63:(111111)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for Heinz numbers of partitions is A300273, counted by A275870.
These compositions are counted by A353860.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A334968 counts distinct sums of subsequences of standard compositions.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A354582 counts distinct partial runs of standard compositions, sums A354907.
Cf. A005811, A029837, A072639, A124771, A274174, A318928, A333381, A353848, A353849, A353850.

repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[repcams[stc[#]],{_}]&]

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

Original entry on oeis.org

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

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.

			The terms together with their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

A357877 The a(n)-th composition in standard order is the sequence of run-sums of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 8, 4, 8, 12, 16, 10, 32, 24, 20, 8, 64, 24, 128, 20, 40, 48, 256, 18, 32, 96, 32, 40, 512, 52, 1024, 16, 80, 192, 72, 40, 2048, 384, 160, 36, 4096, 104, 8192, 80, 68, 768, 16384, 34, 128, 96, 320, 160, 32768, 96, 144, 72, 640, 1536, 65536, 84

Offset: 1

Gus Wiseman, Oct 17 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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).
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.

			The prime indices of 24 are (1,1,1,2), with run-sums (3,2), and this is the 18th composition in standard order, so a(24) = 18.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for prime indices instead of standard compositions is A353832.
The version for standard compositions instead of prime indices is A353847.
A ranking of the rows of A354584.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A056239 adds up prime indices, row sums of A112798.
A066099 lists standard compositions.
A351014 counts distinct runs in standard compositions.
Cf. A118914, A181819, A238279, A239312, A275870, A300273, A304405, A304442, A304660, A333755, A353743-A354912, A357875.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[primeMS[n]]],{n,100}]

cp's OEIS Frontend

A354582 Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.

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A354909 Number of integer compositions of n that are not the run-sums of any other composition.

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A354910 Number of compositions of n that are the run-sums of some other composition.

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A357181 Last run-length of the n-th composition in standard order.

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A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

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A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

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A357877 The a(n)-th composition in standard order is the sequence of run-sums of the prime indices of n.

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