A374761 -id:A374761 - cp's OEIS frontend

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

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

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

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 22, 31, 32, 33, 34, 36, 37, 42, 45, 63, 64, 65, 66, 68, 69, 73, 76, 85, 86, 90, 127, 128, 129, 130, 132, 133, 136, 137, 146, 148, 153, 170, 173, 181, 182

Offset: 1

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 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  21: (2,2,1)
  22: (2,1,2)
  31: (1,1,1,1,1)
  32: (6)
  33: (5,1)
  34: (4,2)
  36: (3,3)
  37: (3,2,1)

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

For leaders of anti-runs we have A374519 (counted by A374517).
For leaders of weakly increasing runs we have A374633, counted by A374631.
The opposite version is A374685 (counted by A374686).
The weak version is A374744.
Compositions of this type are counted by A374760.
For distinct instead of identical runs we have A374767 (counted by A374761).
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
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. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.

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

A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107

Offset: 0

Gus Wiseman, Aug 14 2024

nonn

An anti-run is a sequence with no adjacent equal parts.
These are partitions with no part appearing more than twice and greatest part appearing only once.
Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.

			The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
                (21)  (31)   (32)   (42)   (43)    (53)    (54)
                      (211)  (41)   (51)   (52)    (62)    (63)
                             (311)  (321)  (61)    (71)    (72)
                                    (411)  (322)   (422)   (81)
                                           (421)   (431)   (432)
                                           (511)   (521)   (522)
                                           (3211)  (611)   (531)
                                                   (3221)  (621)
                                                   (4211)  (711)
                                                           (4221)
                                                           (4311)
                                                           (5211)
                                                           (32211)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Includes all strict partitions A000009.
For identical instead of distinct see: A034296, A115029, A374760, A374759.
For compositions instead of partitions we have A374761.
For minima instead of maxima we have A375134, ranks A375398.
The complement is counted by A375401, ranks A375403.
These partitions are ranked by A375402, for compositions A374767.
The complement for minima instead of maxima is A375404, ranks A375399.
A000041 counts integer partitions.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A141199, A279790, A358830, A358833, A358836, A358905, A374704, A374757, A374758, A375136, A375400.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

A_x(N) = {my(x='x+O('x^N), f=sum(i=0,N,(x^i)*prod(j=1,i-1,(1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024

G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 2, 0, 1, 2, 6, 4, 3, 0, 1, 3, 9, 8, 7, 4, 0, 1, 3, 13, 15, 16, 11, 5, 0, 1, 4, 17, 24, 32, 28, 16, 6, 0, 1, 4, 23, 36, 58, 58, 44, 24, 8, 0, 1, 5, 28, 52, 96, 115, 100, 71, 34, 10, 0, 1, 5, 35, 72, 151, 203, 211, 176, 109, 49, 12

Offset: 0

Gus Wiseman, Jul 26 2024

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.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   1   2
   0   1   2   3   2
   0   1   2   6   4   3
   0   1   3   9   8   7   4
   0   1   3  13  15  16  11   5
   0   1   4  17  24  32  28  16   6
   0   1   4  23  36  58  58  44  24   8
   0   1   5  28  52  96 115 100  71  34  10
   0   1   5  35  72 151 203 211 176 109  49  12
Row n = 6 counts the following compositions:
  .  (111111)  (222)    (33)     (42)    (51)    (6)
               (2211)   (321)    (411)   (141)   (15)
               (21111)  (3111)   (132)   (114)   (24)
                        (1221)   (1311)  (312)   (123)
                        (1122)   (1131)  (231)
                        (12111)  (1113)  (213)
                        (11211)  (2121)  (1212)
                        (11121)  (2112)
                        (11112)

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

Column n = k is A000009.
Column k = 2 is A004526.
Row-sums are A011782.
For length instead of sum we have A238343.
The opposite rank statistic is A374630, row-sums of A374629.
Column k = 3 is A374702.
The center n = 2k is A374703.
The corresponding rank statistic is A374741 row-sums of A374740.
Types of runs (instead of weakly decreasing):
- For leaders of constant runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of strictly decreasing runs we have A374766.
Types of run-leaders:
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742, ranks A374744.
- For distinct leaders we have A374743, ranks A374701.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
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. A000041, A106356, A124766, A261982, A333213, A374251, A374761.

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

A375126 Strictly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 5, 12, 6, 14, 15, 16, 8, 4, 9, 20, 10, 10, 11, 24, 12, 26, 13, 28, 14, 30, 31, 32, 16, 8, 17, 36, 4, 18, 19, 40, 20, 42, 21, 20, 10, 22, 23, 48, 24, 12, 25, 52, 26, 26, 27, 56, 28, 58, 29, 60, 30, 62, 63, 64, 32, 16, 33, 8, 8

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly decreasing runs in the n-th composition in standard order.
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 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.
Does this sequence contain all nonnegative integers?

			The 813th composition in standard order is (1,3,2,1,2,1), with strictly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

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 A374767, counted by A374761.
Positions of elements of A272919 are A374759, counted by A374760.
Ranks of rows of A374757 (row-sums A374758).
The weak opposite version is A375123.
The weak version is A375124.
The opposite version is A375125.
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) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
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. A065120, A106356, A188920, A238343, A333213, A373949, A374634, A374685, A374744, A374766, 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],Greater]],{n,0,100}]

A000120(a(n)) = A124769(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374758(n).

A374758 Sum of leaders of strictly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 5, 6, 3, 5, 5, 6, 5, 6, 5, 5, 4, 5, 5, 6, 5, 4, 5, 6, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 7, 6, 5, 6, 4, 4, 6, 6, 7, 6, 5, 4, 6, 5, 6, 6, 7, 6, 5, 6, 7, 6, 6

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 maximal strictly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)) with leaders (3,2,2,2,5,1,1), so a(1234567) = 16.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Row sums of A374757.
For leaders of constant runs we have A373953.
For leaders of anti-runs we have A374516.
For leaders of weakly increasing runs we have A374630.
For length instead of sum we have A124769.
The opposite version is A374684, sum of A374683 (length A124768).
The case of partitions ranked by Heinz numbers is A374706.
The weak version is A374741, sum of A374740 (length A124765).
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-compression transform is A373948.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
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. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, A374700, A374759, A374761, A374767.

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

A375134 Number of integer partitions of n whose maximal anti-runs have distinct minima.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 18, 21, 28, 33, 43, 52, 66, 78, 98, 116, 145, 171, 209, 247, 300, 352, 424, 499, 595, 695, 826, 963, 1138, 1322, 1553, 1802, 2106, 2435, 2835, 3271, 3795, 4365, 5046, 5792, 6673, 7641, 8778, 10030, 11490, 13099, 14968, 17030

Offset: 0

Gus Wiseman, Aug 14 2024

nonn

These are partitions with no part appearing more than twice and with the least part appearing only once.

Also the number of reversed integer partitions of n whose maximal anti-runs have distinct minima.

			The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with minima (5,3,1), so y is counted under a(29).
The a(1) = 1 through a(9) = 11 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (12)  (13)  (14)   (15)   (16)   (17)    (18)
                        (23)   (24)   (25)   (26)    (27)
                        (122)  (123)  (34)   (35)    (36)
                                      (124)  (125)   (45)
                                      (133)  (134)   (126)
                                             (233)   (135)
                                             (1223)  (144)
                                                     (234)
                                                     (1224)
                                                     (1233)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Includes all strict partitions A000009.
For identical instead of distinct leaders we have A115029.
A version for compositions instead of partitions is A374518, ranks A374638.
For minima instead of maxima we have A375133, ranks A375402.
These partitions have ranks A375398.
The complement is counted by A375404, ranks A375399.
A000041 counts integer partitions.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A141199, A358830, A358836, A358905, A374704, A374757, A374758, A374761, A375136, A375400, A375401.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]

A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,(x^i)*prod(j=i+1,N-i,(1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024

G.f.: 1 + Sum_{i>0} (x^i * Product_{j>i} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375126 Strictly decreasing run-leader transformation for standard compositions.

Views

Author