A188920 -id:A188920 - cp's OEIS frontend

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Gus Wiseman, Aug 06 2024

nonn

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

			The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

For leaders of constant runs we have A056823.
For leaders of weakly increasing runs we have A374636, complement A189076?
The complement is counted by A374697.
For leaders of anti-runs we have A374699, complement A374682.
Other functional neighbors: A188920, A374764, A374765.
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.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

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

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

Original entry on oeis.org

13, 25, 27, 29, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  82: (2,3,2)
  89: (2,1,3,1)
  91: (2,1,2,1,1)
  93: (2,1,1,2,1)

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

For leaders of identical runs we have A335485.
Positions of non-strictly decreasing rows in A374629 (sums A374630).
For identical leaders we have A374633, counted by A374631.
Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.
Interchanging weak/strict gives A375139, counted by A375135.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375296.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly increasing 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[stc[#],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x<=y

A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

Original entry on oeis.org

13, 25, 27, 29, 41, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177, 179, 181, 182

Offset: 1

Gus Wiseman, Aug 13 2024

nonn

The leaders of maximal 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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 23-1 or 12-1.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)

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

For leaders of identical runs we have A335486, reverse A335485.
Matching 1-32 only gives A375138, reverse A375137, both counted by A374636.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375295.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly increasing 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A333213, A335480, A335482, A373948, A374630, A374633, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[Reverse[stc[#]],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x<=y

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 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 terms together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

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

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing 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.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

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

A375297 Number of integer compositions of n matching both of the dashed patterns 23-1 and 1-32.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 21, 68, 199, 545, 1410, 3530, 8557, 20255, 46968, 107135, 240927, 535379, 1177435, 2566618, 5551456

Offset: 0

Gus Wiseman, Aug 23 2024

Also the number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing and whose reverse satisfies the same condition.

			The a(0) = 0 through a(11) = 21 compositions:
  .  .  .  .  .  .  .  .  .  (12321)  (1342)    (1352)
                                      (2431)    (2531)
                                      (12421)   (11342)
                                      (13231)   (12431)
                                      (112321)  (12521)
                                      (123211)  (13241)
                                                (13421)
                                                (14231)
                                                (23132)
                                                (24311)
                                                (112421)
                                                (113231)
                                                (122321)
                                                (123212)
                                                (123221)
                                                (124211)
                                                (132311)
                                                (212321)
                                                (1112321)
                                                (1123211)
                                                (1232111)

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

For leaders of identical runs we have A332834.
For just one of the two conditions we have A374636, ranks A375137/A375138.
These compositions are ranked by A375407.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
Cf. A000041, A056823, A188920, A189076, A238343, A333213, A335514, A374631, A374632, A374635, A374681.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,y_,z_,_,x_,_}/;x_,x_,_,z_,y_,_}/;x

A375406 Number of integer compositions of n that match the dashed pattern 3-12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 14, 41, 110, 278, 673, 1576, 3599, 8055, 17732, 38509, 82683, 175830, 370856, 776723, 1616945, 3348500, 6902905, 14174198, 29004911, 59175625, 120414435, 244468774, 495340191, 1001911626, 2023473267, 4081241473, 8222198324, 16548146045, 33276169507

Offset: 0

Gus Wiseman, Aug 22 2024

nonn

First differs from the non-dashed version A335514 at a(9) = 41, A335514(9) = 42, due to the composition (3,1,3,2).

Also the number of integer compositions of n whose leaders of weakly decreasing runs are not 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 not counted under a(13); also q does not match 3-12. On the other hand, the reverse is (3,1,2,2,1,2,1,1), with maximal weakly decreasing runs ((3,1),(2,2,1),(2,1,1)), with leaders (3,2,2), which are not weakly increasing, so it is counted under a(13); meanwhile it matches 3-12, as required.

			The a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  .  (312)  (412)   (413)
                           (1312)  (512)
                           (3112)  (1412)
                           (3121)  (2312)
                                   (3122)
                                   (3212)
                                   (4112)
                                   (4121)
                                   (11312)
                                   (13112)
                                   (13121)
                                   (31112)
                                   (31121)
                                   (31211)

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

For leaders of identical runs we have A056823.
The complement is counted by A188900.
The non-dashed version is A335514, ranks A335479.
Ranks are positions of non-weakly increasing rows in A374740.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
Counting compositions by number of runs: A238130, A238279, A333755.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188920, A189076, A189077, A238343, A333213, A335548, A374629, A374637, A374679, A374748.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,z_,_,x_,y_,_}/;x

a(n>0) = 2^(n-1) - A188900(n).

A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 77, 96, 126, 167, 217, 278, 402, 488, 647, 822, 1073, 1340, 1747, 2324, 2890, 3695, 4690, 5924, 7469, 9407, 11718, 15405, 18794, 23777, 29507, 37188, 45720, 57404, 70358, 87596, 110672, 135329, 167018, 206761, 254200, 311920

Offset: 0

Gus Wiseman, Sep 18 2024

nonn

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

			The a(1) = 1 through a(9) = 11 compositions:
 (1) (2) (3)   (4)   (5)   (6)     (7)     (8)     (9)
         (1,2) (1,3) (1,4) (1,5)   (1,6)   (1,7)   (1,8)
                     (2,3) (2,4)   (2,5)   (2,6)   (2,7)
                           (1,2,3) (3,4)   (3,5)   (3,6)
                           (1,3,2) (1,2,4) (1,2,5) (4,5)
                                   (1,4,2) (1,3,4) (1,2,6)
                                           (1,4,3) (1,3,5)
                                           (1,5,2) (1,5,3)
                                                   (1,6,2)
                                                   (2,3,4)
                                                   (2,4,3)

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 less-greater or greater-less we have A294617.
This is a strict case of A374688, weak version A374635.
The strict less-greater version is A374689, weak version A189076.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions, strict A032020.
A238130, A238279, A333755 count compositions by number of runs.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000110, A008289, A056823, A106356, A188920, A238343, A261982, A274174, A333213, A374634, A374683, A374698, A374763.

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

\\ here Q(n) gives n-th row of A008289.
Q(n)={Vecrev(polcoef(prod(k=1, n, 1 + y*x^k, 1 + O(x*x^n)), n)/y)}
a(n)={if(n==0, 1, my(r=Q(n), s=Vec(serlaplace(exp(exp(x+O(x^#r))- 1)))); sum(k=1, #r, r[k]*s[k]))} \\ Andrew Howroyd, Sep 18 2024

a(n) = Sum_{k>=1} A008289(n,k)*A000110(k-1) for n > 0. - Andrew Howroyd, Sep 18 2024

a(26) onwards from Andrew Howroyd, Sep 18 2024

A188929 Continued fraction expansion of sqrt(12)+sqrt(13).

Original entry on oeis.org

7, 14, 2, 1, 4, 21, 3, 9, 1, 4, 2, 1, 1, 1, 2, 3, 25, 1, 4, 2, 1, 28, 1, 2, 1, 1, 33, 1, 1, 2, 4, 5, 17, 2, 1, 1, 1, 2, 2, 1, 5, 1, 13, 3, 1, 1, 1, 1, 1, 2, 1, 1, 34, 1, 3, 4, 7, 1, 62, 1, 38, 1, 6, 1, 2, 1, 1, 2, 3, 1, 2, 2, 196, 1, 1, 1, 26, 1, 2, 1, 2, 2, 1, 16, 1, 15, 5, 1, 1, 2, 12, 3, 1, 2, 2, 1, 2, 1, 2, 2, 12, 5, 2, 5, 17, 1, 5, 1, 1, 36, 2, 1, 5, 1, 8, 3, 50, 7, 7, 3

Offset: 0

Clark Kimberling, Apr 13 2011

For a geometric interpretation, see A188640 and A188928.

			sqrt(12)+sqrt(13)=[7,14,2,1,4,21,3,9,1,4,2,1,1,1,...].

Cf. A188920, A188928 (decimal expansion).

r = 48^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Offset changed by Andrew Howroyd, Jul 08 2024

cp's OEIS Frontend

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375297 Number of integer compositions of n matching both of the dashed patterns 23-1 and 1-32.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375406 Number of integer compositions of n that match the dashed pattern 3-12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A188929 Continued fraction expansion of sqrt(12)+sqrt(13).

Views

Author