A374758 -id:A374758 - cp's OEIS frontend

A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

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

Offset: 0

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly 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.

			The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) is 3+2+2+5 = 12.

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

For length instead of sum we have A124765.
Other types of runs are A373953, A374516, A374684, A374758.
The opposite is A374630.
Row-sums of A374740, opposite A374629.
Counting compositions by this statistic gives A374748, opposite A374637.
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.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564, counted by A032020.
- Constant compositions are ranked by A272919.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948.
Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A373949, A373954, A374631, A374633, A374635, A374701, A374742.

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

A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 26 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 maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)) with leaders (3,2,1,2,1,1,1,1), so a(1234567) = 12.

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

The weak version is A374630.
Row-sums of A374683.
The opposite version is A374758.
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.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Run-length transform is A333627.
- Run-compression transform is A373948.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.
Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).

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

A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 31 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with maximal anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) is 3 + 2 + 1 + 1 = 7.

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

For length instead of sum we have A333381.
Row-sums of A374515.
Other types of runs (instead of anti-):
- For identical runs we have A373953, row-sums of A374251.
- For weakly increasing runs we have A374630, row-sums of A374629.
- For strictly increasing runs we have A374684, row-sums of A374683.
- For weakly decreasing runs we have A374741, row-sums of A374740.
- For strictly decreasing runs we have A374758, row-sums of A374757.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
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.
- Anti-runs are ranked by A333489, counted by A003242.
- 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 maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A272919, A373949, A374517, A374518, A374519, A374638.

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

A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 8, 7, 0, 0, 0, 1, 3, 17, 11, 0, 0, 0, 0, 4, 10, 35, 15, 0, 0, 0, 0, 1, 12, 28, 65, 22, 0, 0, 0, 0, 1, 6, 31, 70, 118, 30, 0, 0, 0, 0, 1, 3, 22, 78, 163, 203, 42, 0, 0, 0, 0, 0, 4, 13, 69, 186, 354, 342, 56

Offset: 0

Gus Wiseman, Aug 02 2024

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.

Are the column-sums finite?

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   0   3   5
   0   0   0   1   8   7
   0   0   0   1   3  17  11
   0   0   0   0   4  10  35  15
   0   0   0   0   1  12  28  65  22
   0   0   0   0   1   6  31  70 118  30
   0   0   0   0   1   3  22  78 163 203  42
   0   0   0   0   0   4  13  69 186 354 342  56
Row n = 6 counts the following compositions:
  .  .  .  (321)  (42)    (51)     (6)
                  (132)   (411)    (15)
                  (2121)  (141)    (24)
                          (312)    (114)
                          (231)    (33)
                          (213)    (123)
                          (3111)   (1113)
                          (1311)   (222)
                          (1131)   (1122)
                          (2211)   (11112)
                          (2112)   (111111)
                          (1221)
                          (1212)
                          (21111)
                          (12111)
                          (11211)
                          (11121)

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

Column n = k is A000041.
Row-sums are A011782.
For length instead of sum we have A333213.
The corresponding rank statistic is A374758, row-sums of A374757.
For identical leaders we have A374760, ranks A374759.
For distinct leaders we have A374761, ranks A374767.
Other types of runs (instead of strictly decreasing):
- For leaders of identical 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 weakly decreasing runs we have A374748.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

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

cp's OEIS Frontend

A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica