A374741 -id:A374741 - cp's OEIS frontend

A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

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

Offset: 0

Gus Wiseman, Jul 24 2024

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 row 1234567 is (3,2,2,5).
The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:
    0: () -> ()           15: (1,1,1,1) -> (1)
    1: (1) -> (1)         16: (5) -> (5)
    2: (2) -> (2)         17: (4,1) -> (4)
    3: (1,1) -> (1)       18: (3,2) -> (3)
    4: (3) -> (3)         19: (3,1,1) -> (3)
    5: (2,1) -> (2)       20: (2,3) -> (2,3)
    6: (1,2) -> (1,2)     21: (2,2,1) -> (2)
    7: (1,1,1) -> (1)     22: (2,1,2) -> (2,2)
    8: (4) -> (4)         23: (2,1,1,1) -> (2)
    9: (3,1) -> (3)       24: (1,4) -> (1,4)
   10: (2,2) -> (2)       25: (1,3,1) -> (1,3)
   11: (2,1,1) -> (2)     26: (1,2,2) -> (1,2)
   12: (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2)
   13: (1,2,1) -> (1,2)   28: (1,1,3) -> (1,3)
   14: (1,1,2) -> (1,2)   29: (1,1,2,1) -> (1,2)

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

Row-leaders are A065120.
Row-lengths are A124765.
Other types of runs are A374251, A374515, A374683, A374757.
The opposite is A374629.
Positions of distinct (strict) rows are A374701, counted by A374743.
Row-sums are A374741, opposite A374630.
Positions of identical rows are A374744, counted by A374742.
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 anti-run compositions are 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.
Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637.

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

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402

Offset: 0

Gus Wiseman, Aug 01 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 a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

John Tyler Rascoe, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296 or A115029.
These compositions have ranks A374519.
The complement is counted by A374640.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly decreasing runs we have A374742, ranks A374741.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
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.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.

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

C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

a(26) onwards from John Tyler Rascoe, Aug 16 2024

A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 20 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 1234567th composition in standard order are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 8.

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

For length instead of sum we have A124766.
For leaders of constant runs we have A373953, excess A373954.
For leaders of anti-runs we have A374516.
Row-sums of A374629.
Counting compositions by this statistic gives A374637.
For leaders of strictly increasing runs we have A374684.
For leaders of weakly decreasing runs we have A374741.
For leaders of strictly decreasing runs we have A374758
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.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Sum is A029837 (or sometimes A070939).
- Listed by A066099.
- Length is A070939.
- 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, A374631, A374633, A374635.

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

A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 29 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.

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 strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2,3)
    6:   (1,2) -> (1,2)     21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2,2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1,4)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1,3)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2,2)
   12:   (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2,1)
   13: (1,2,1) -> (1,2)     28:   (1,1,3) -> (1,1,3)
   14: (1,1,2) -> (1,1,2)   29: (1,1,2,1) -> (1,1,2)

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

Row-leaders of nonempty rows are A065120.
Row-lengths are A124769.
The opposite version is A374683, sum A374684, length A124768.
The weak version is A374740, sum A374741, length A124765.
Row-sums are A374758.
Positions of identical rows are A374759 (counted by A374760).
Positions of distinct (strict) rows are 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. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.

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

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 87, 138, 220, 349, 556, 881, 1403, 2229, 3551, 5653, 9019, 14387, 22988, 36739, 58785, 94100, 150765, 241658, 387617, 622002, 998658, 1604032, 2577512, 4143243, 6662520, 10716931, 17243904, 27753518, 44680121, 71947123, 115880662

Offset: 0

Gus Wiseman, Jul 25 2024

nonn

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

			The composition (3,1,3,2,1,3,3) has maximal weakly decreasing subsequences ((3,1),(3,2,1),(3,3)), with leaders (3,3,3), so is counted under a(16).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (211)   (212)    (51)
                        (1111)  (221)    (222)
                                (311)    (321)
                                (2111)   (411)
                                (11111)  (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374744 = positions of identical rows in A374740, cf. A374629.
Types of runs (instead of weakly decreasing):
- 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 strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Types of run-leaders (instead of identical):
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
- For distinct leaders we have A374743, ranks A374701.
- For weakly increasing leaders we appear to have A188900.
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.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374741.

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

B(i) = x^i/(1-x^i) * sum(j=1,i-1, x^j*prod(k=1,j, (1-x^k)^(-1)))
A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N,-1+(1+x^i/(1-x^i))/(1-B(i))))}
A_x(30) \\ John Tyler Rascoe, Apr 29 2025

G.f.: 1 + Sum_{i>0} -1 + (1 + x^i/(1 - x^i))/(1 - B(i,x)) where B(i,x) = x^i/(1 - x^i) * Sum_{j=1..i-1} x^j * Product_{k=1..j} (1 - x^k)^(-1). - John Tyler Rascoe, Apr 29 2025

a(24)-a(40) from Alois P. Heinz, Jul 26 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

A188900 Number of compositions of n that avoid the pattern 12-3.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 60, 114, 215, 402, 746, 1375, 2520, 4593, 8329, 15036, 27027, 48389, 86314, 153432, 271853, 480207, 845804, 1485703, 2603018, 4549521, 7933239, 13803293, 23966682, 41530721, 71830198, 124010381, 213725823, 367736268, 631723139, 1083568861

Offset: 0

Nathaniel Johnston, Apr 17 2011

nonn

First differs from the non-dashed version A102726 at a(9) = 215, A102726(9) = 214, due to the composition (1,3,2,3).
The value a(11) = 7464 in Heubach et al. is a typo.
Theorem: A composition avoids 3-12 iff its leaders of maximal weakly decreasing runs are weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is counted under a(13); also q avoids 3-12, as required. On the other hand, the composition q = (3,2,1,2,2,1,2) has maximal weakly decreasing runs ((3,2,1),(2,2,1),(2)), with leaders (3,2,2), which are not weakly increasing, so q is not counted under a(13); also q matches 3-12, as required. - Gus Wiseman, Aug 21 2024

			The initial terms are too dense, but see A375406 for the complement. - _Gus Wiseman_, Aug 21 2024

Nathaniel Johnston, Table of n, a(n) for n = 0..200
S. Heubach, T. Mansour and A. O. Munagi, Avoiding Permutation Patterns of Type (2,1) in Compositions, Online Journal of Analytic Combinatorics, 4 (2009).

The non-dashed version A102726, non-ranks A335483.
For 23-1 we have A189076.
The non-ranks are a subset of A335479 and do not include 404, 788, 809, ...
For strictly increasing leaders we have A358836, ranks A326533.
The strict version is A374762.
The complement is counted by A375406.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A188920, A238343, A261982, A333175, A333213, A374630, A374679, A374688, A374740, A374741.

with(PolynomialTools):n:=20:taypoly:=taylor(mul(1/(1 - x^i/mul(1-x^j,j=1..i-1)),i=1..n),x=0,n+1):seq(coeff(taypoly,x,m),m=0..n);

m = 35;
Product[1/(1 - x^i/Product[1 - x^j, {j, 1, i - 1}]), {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Mar 31 2020 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

G.f.: Product_{i>=1} (1/(1 - x^i/Product_{j=1..i-1} (1 - x^j))).

a(n) = 2^(n-1) - A375406(n). - Gus Wiseman, Aug 22 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}]

A375124 Weakly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 8, 4, 2, 2, 12, 6, 6, 1, 16, 8, 4, 4, 20, 2, 10, 2, 24, 12, 6, 6, 12, 6, 6, 1, 32, 16, 8, 8, 4, 4, 18, 4, 40, 20, 2, 2, 20, 10, 10, 2, 48, 24, 12, 12, 52, 6, 26, 6, 24, 12, 6, 6, 12, 6, 6, 1, 64, 32, 16, 16, 8, 8, 34, 8, 72, 4, 4, 4, 36

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly decreasing runs in the n-th composition in standard order.
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 813th composition in standard order is (1,3,2,1,2,1), with weakly 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 A374701, counted by A374743.
Positions of elements of A272919 are A374744, counted by A374742.
Ranks of rows of A374740.
The opposite version is A375123.
The strict version is A375126.
The strict 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, sum A070939.
- 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, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.

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],GreaterEqual]],{n,0,100}]

A000120(a(n)) = A124765(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374741(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}]

cp's OEIS Frontend

A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

A188900 Number of compositions of n that avoid the pattern 12-3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula