A333213 -id:A333213 - cp's OEIS frontend

A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 27, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 99, 102, 103, 104, 105, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Jul 27 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

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 maximal strictly increasing subsequences of the 6560th composition in standard order are ((1,3),(1,2,6)), with leaders (1,1), so 6560 is in the sequence.
The terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The weak version is A374633, counted by A374631.
Positions of constant rows in A374683.
Compositions of this type are counted by A374686.
For distinct leaders we have A374698, counted by A374687.
The opposite version is A374759, counted by A374760.
Other types of runs: A272919 (counts A000005), A374519 (counts A374517), A374744 (counts A374742).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374701, A374740, A374768.

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

A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 48, 67, 99, 133, 185, 245, 333, 432, 574, 732, 957, 1208, 1554, 1941, 2468, 3060, 3844, 4731, 5893, 7204, 8898, 10816, 13268, 16043, 19546, 23523, 28497, 34150, 41147, 49106, 58892, 70020, 83597, 99047, 117778, 139087

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of non-weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(6) = 1 through a(10) = 13 partitions:
  (321)  (421)   (431)    (432)     (532)
         (3211)  (521)    (531)     (541)
                 (4211)   (621)     (631)
                 (32111)  (3321)    (721)
                          (4311)    (4321)
                          (5211)    (5311)
                          (42111)   (6211)
                          (321111)  (32221)
                                    (33211)
                                    (43111)
                                    (52111)
                                    (421111)
                                    (3211111)

The strong version for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
The complement is counted by A349060.
These partitions are ranked by A349794.
The non-strict case is A349796, complement A349795.
The strong case is A349801.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating compositions, ranked by A345167.
A025048 and A025049 count directed alternating compositions.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349056 counts weakly alternating permutations of prime indices.
A349798 counts weakly but not strongly alternating perms of prime indices.
Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.

Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 29, 51, 91, 162, 291, 523, 948, 1712, 3112, 5656, 10297, 18763, 34217, 62442, 114006, 208239, 380465, 695342, 1271046, 2323818, 4249113, 7770389, 14210991, 25991853, 47541734, 86962675, 159077005, 291001483, 532345978, 973871397

Offset: 0

Gus Wiseman, Jul 27 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are identical. For maxima instead of minima we have A374760. For all partitions (not just strict) we have A374704, for maxima A358905.

			The composition (2,3,2,2,3,4) has strictly increasing runs ((2,3),(2),(2,3,4)), with leaders (2,2,2), so is counted under a(16).
The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (131)    (114)
                        (1111)  (1112)   (123)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1131)
                                         (1212)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374685.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374517, ranks A374519.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of weakly decreasing runs we have A374742, ranks A374744.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Types of run-leaders (instead of identical):
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- For weakly increasing leaders we have A374690.
- For weakly decreasing leaders we have A374697.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374683 lists leaders of strictly increasing runs of standard compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A304969, A333213, A374632, A374634, A374635, A374640.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@First/@Split[#,Less]&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 27 2024

a(26) onwards from Andrew Howroyd, Jul 27 2024

A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 105, 198, 371, 690, 1280, 2364, 4353, 7981, 14568, 26466, 47876, 86264, 154896, 277236, 494675, 879924, 1560275, 2757830, 4859010, 8534420, 14945107, 26096824, 45446624, 78939432, 136773519, 236401194, 407614349, 701147189, 1203194421

Offset: 0

Gus Wiseman, Jul 25 2024

nonn

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The composition (1,3,1,4,1,2,2,1) has maximal weakly decreasing subsequences ((1),(3,1),(4,1),(2,2,1)), with leaders (1,3,4,2), so is counted under a(15).
The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374701 = positions of distinct rows in A374740, opposite A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of anti-runs we have A374518, ranks A374638.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742.
- For strictly increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A188920, A189076, A238343, A261982, A333213, A374635, A374741.

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

a(24)-a(36) from Alois P. Heinz, Jul 26 2024

A374633 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 26, 27, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115

Offset: 1

Gus Wiseman, Jul 21 2024

nonn

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

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 maximal weakly increasing subsequences of the 26165th composition in standard order are ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so 26165 is in the sequence.
The sequence together with the corresponding compositions begins:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For strictly decreasing leaders we appear to have A188920.
For weakly decreasing leaders we appear to have A189076.
Other types of runs: A272919 (counted by A000005), A374519 (counted by A374517), A374685 (counted by A374686), A374744 (counted by A374742), A374759 (counted by A374760).
Positions of constant rows in A374629 (which has sums A374630).
Compositions of this type are counted by A374631.
For strictly increasing leaders see A374634.
For all different leaders we have A374768, counted by A374632.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374637 counts compositions by sum of leaders of weakly increasing runs.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Sum is A029837 (or sometimes A070939).
- Parts are listed by A066099.
- Length is A070939.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333175, A333213, A373949, A374635.

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

A374637 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly increasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 2, 0, 3, 2, 1, 2, 0, 5, 4, 3, 1, 3, 0, 7, 10, 7, 3, 1, 4, 0, 11, 19, 14, 9, 4, 2, 5, 0, 15, 39, 27, 22, 10, 7, 2, 6, 0, 22, 69, 59, 48, 24, 15, 8, 3, 8, 0, 30, 125, 117, 104, 56, 38, 19, 10, 3, 10, 0, 42, 211, 241, 215, 132, 80, 49, 25, 12, 5, 12

Offset: 0

Gus Wiseman, Jul 23 2024

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   0   2
   0   3   2   1   2
   0   5   4   3   1   3
   0   7  10   7   3   1   4
   0  11  19  14   9   4   2   5
   0  15  39  27  22  10   7   2   6
   0  22  69  59  48  24  15   8   3   8
   0  30 125 117 104  56  38  19  10   3  10
   0  42 211 241 215 132  80  49  25  12   5  12
   0  56 354 473 445 296 186 109  61  31  17   5  15
   0  77 571 917 896 665 409 258 139  78  41  20   7  18
Row n = 6 counts the following compositions:
  .  (15)      (24)     (33)     (312)   (411)  (6)
     (114)     (141)    (231)    (3111)         (51)
     (123)     (1311)   (213)    (2121)         (42)
     (1113)    (1131)   (132)                   (321)
     (1122)    (222)    (2211)
     (11112)   (1221)   (2112)
     (111111)  (1212)   (21111)
               (12111)
               (11211)
               (11121)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Last column n = k is A000009.
Second column k = 2 is A000041.
Row-sums are A011782.
For length instead of sum we have A238343.
The corresponding rank statistic is A374630, row-sums of A374629.
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
Types of run-leaders:
- For strictly decreasing leaders we appear to have A188920.
- For weakly decreasing leaders we appear to have A189076.
- For identical leaders we have A374631.
- For distinct leaders we have A374632, ranks A374768.
- For strictly increasing leaders we have A374634.
- For weakly increasing leaders we have A374635.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
Cf. A106356, A124766, A261982, A333213, A374251, A374518, A374687, A374761.

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

A374701 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

First differs from A335469 in having 150, which corresponds to the composition (3,2,1,2).

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 maximal weakly decreasing subsequences of the 1257th composition in standard order are ((3,1,1),(2),(3,1)), with leaders (3,2,3), so 1257 is not in the sequence.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of distinct (strict) rows in A374740, opposite A374629.
Compositions of this type are counted by A374743.
For identical leaders we have A374744, counted by A374742.
Other types of runs and their counts: A374249 (A274174), A374638 (A374518), A374698 (A374687), A374767 (A374761), A374768 (A374632).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
Cf. A065120, A106356, A189076, A238343, A261982, A272919, A333213, A374630, A374635, A374637, A374748.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#],GreaterEqual]&] (* Gus Wiseman, Jul 24 2024 *)

A374760 Number of integer compositions of n whose leaders of strictly decreasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 15, 21, 28, 38, 52, 70, 95, 129, 173, 234, 318, 428, 579, 784, 1059, 1433, 1942, 2630, 3564, 4835, 6559, 8902, 12094, 16432, 22340, 30392, 41356, 56304, 76692, 104499, 142448, 194264, 265015, 361664, 493749, 674278, 921113, 1258717

Offset: 0

Gus Wiseman, Jul 29 2024

nonn

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

			The composition (3,3,2,1,3,2,1) has strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so is counted under a(15).
The a(0) = 1 through a(8) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (21)   (22)    (32)     (33)      (43)       (44)
                 (111)  (31)    (41)     (42)      (52)       (53)
                        (1111)  (212)    (51)      (61)       (62)
                                (221)    (222)     (313)      (71)
                                (11111)  (321)     (331)      (323)
                                         (2121)    (421)      (332)
                                         (111111)  (2122)     (431)
                                                   (2212)     (521)
                                                   (2221)     (2222)
                                                   (1111111)  (3131)
                                                              (21212)
                                                              (21221)
                                                              (22121)
                                                              (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296.
The weak version is A374742, ranks A374744.
The opposite version is A374686, ranks A374685.
The weak opposite version is A374631, ranks A374633.
Ranked by A374759.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374517, ranks A374519.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374640, A374761.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@First/@Split[#,Greater]&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 31 2024

G.f.: 1 + Sum_{k>=1} -1 + 1/(1 - x^k*Product_{j=1..k-1} (1 + x^j)). - Andrew Howroyd, Jul 31 2024

a(24) onwards from Andrew Howroyd, Jul 31 2024

A375123 Weakly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 1, 1, 8, 9, 2, 5, 1, 3, 1, 1, 16, 17, 18, 9, 2, 5, 5, 5, 1, 3, 1, 3, 1, 3, 1, 1, 32, 33, 34, 17, 4, 37, 9, 9, 2, 5, 2, 5, 5, 11, 5, 5, 1, 3, 6, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 64, 65, 66, 33, 68, 69, 17, 17, 4, 9, 18, 37, 9, 19, 9, 9

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
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 813th composition in standard order is (1,3,2,1,2,1), with weakly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

Gus Wiseman, Statistics, classes, and transformations of standard compositions.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of elements of A233564 are A374768, counted by A374632.
Positions of elements of A272919 are A374633, counted by A374631.
Ranks of rows of A374629.
The opposite version is A375124.
The strict version is A375125.
The strict opposite version is A375126.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-sum transformation is A353847.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A189076, A238343, A333213, A373949, A374634, A374635, A374637, A374743, A374744, A375128.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],LessEqual]],{n,0,100}]

A000120(a(n)) = A124766(n).
A070939(a(n)) = A374630(n) for n > 0.
A065120(a(n)) = A065120(n).

A374706 Sum of minima of the maximal strictly increasing runs in the weakly increasing prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 3, 8, 2, 2, 1, 9, 3, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 4, 12, 1, 2, 3, 13, 1, 14, 2, 4, 1, 15, 4, 8, 4, 2, 2, 16, 5, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 5, 21, 1, 5, 2, 4, 1, 22, 4, 8, 1

Offset: 1

Gus Wiseman, Aug 04 2024

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 prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), summing to a(540) = 6.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of constant runs we have A066328.
A version for compositions is A374684, row-sums of A374683 (length A124768).
Row-sums of A375128.
For length instead of sum we have A375136.
A055887 counts sequences of partitions with total sum n.
A112798 lists prime indices:
- length A001222, distinct A001221
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A141199, A189076, A218482, A279790, A333213, A358836, A374634, A374700, A374758, A375133.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Split[prix[n],Less]],{n,100}]

cp's OEIS Frontend

A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

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A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.

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A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

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A374633 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are identical.

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A374637 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly increasing runs sum to k.

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A374701 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are distinct.

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A374760 Number of integer compositions of n whose leaders of strictly decreasing runs are identical.

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