Displaying 1-10 of 21 results found.
Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.
+10
87
1, 2, 1, 3, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 3, 3, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1
COMMENTS
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.
EXAMPLE
The 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4,1)
3: (1,1) -> (1) 18: (3,2) -> (3,2)
4: (3) -> (3) 19: (3,1,1) -> (3,1)
5: (2,1) -> (2,1) 20: (2,3) -> (2)
6: (1,2) -> (1) 21: (2,2,1) -> (2,1)
7: (1,1,1) -> (1) 22: (2,1,2) -> (2,1)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1)
9: (3,1) -> (3,1) 24: (1,4) -> (1)
10: (2,2) -> (2) 25: (1,3,1) -> (1,1)
11: (2,1,1) -> (2,1) 26: (1,2,2) -> (1)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1)
13: (1,2,1) -> (1,1) 28: (1,1,3) -> (1)
14: (1,1,2) -> (1) 29: (1,1,2,1) -> (1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], LessEqual], {n, 0, 100}]
CROSSREFS
Positions of non-weakly decreasing rows are A375137. A335456 counts patterns matched by compositions.
All of the following pertain to compositions in standard order:
- Ranks of non-contiguous compositions are A374253, counted by A335548. Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.
Numbers k such that the k-th composition in standard order has its equal parts contiguous.
+10
70
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85
COMMENTS
These are compositions avoiding the patterns (1,2,1) and (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.
EXAMPLE
The terms together with their standard compositions begin:
0: ()
1: (1)
2: (2)
3: (1,1)
4: (3)
5: (2,1)
6: (1,2)
7: (1,1,1)
8: (4)
9: (3,1)
10: (2,2)
11: (2,1,1)
12: (1,3)
14: (1,1,2)
15: (1,1,1,1)
16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@First/@Split[stc[#]]&]
CROSSREFS
Compositions of this type are counted by A274174. Permutations of prime indices of this type are counted by A333175. A066099 lists compositions in standard order.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.
Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.
+10
42
1, 1, 2, 3, 6, 10, 20, 36, 69, 130, 247, 467, 890, 1689, 3213, 6110, 11627, 22121, 42101, 80124, 152512, 290300, 552609, 1051953, 2002583, 3812326
COMMENTS
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.
EXAMPLE
The composition (1,3,3,2,4,2) has weakly increasing runs ((1,3,3),(2,4),(2)), with leaders (1,2,2), so is counted under a(15).
The a(0) = 1 through a(6) = 20 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(121) (122) (114)
(1111) (131) (123)
(1112) (132)
(1121) (141)
(1211) (222)
(11111) (1113)
(1122)
(1131)
(1212)
(1221)
(1311)
(11112)
(11121)
(11211)
(12111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], LessEqual@@First/@Split[#, LessEqual]&]], {n, 0, 15}]
CROSSREFS
Ranked by positions of weakly increasing rows in A374629 (sums A374630). Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A000041. - For leaders of weakly decreasing runs we have A188900. - For leaders of anti-runs we have A374681. - For leaders of strictly increasing runs we have A374690. - For leaders of strictly decreasing runs we have A374764. Types of run-leaders (instead of weakly increasing):
- 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 strictly increasing leaders we have A374634. A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.
+10
41
1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 1, 3, 1, 1, 1
COMMENTS
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.
EXAMPLE
The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so row 1234567 is (3,2,1,2,1,1,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly increasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1,1,1,1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4,1)
3: (1,1) -> (1,1) 18: (3,2) -> (3,2)
4: (3) -> (3) 19: (3,1,1) -> (3,1,1)
5: (2,1) -> (2,1) 20: (2,3) -> (2)
6: (1,2) -> (1) 21: (2,2,1) -> (2,2,1)
7: (1,1,1) -> (1,1,1) 22: (2,1,2) -> (2,1)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1,1,1)
9: (3,1) -> (3,1) 24: (1,4) -> (1)
10: (2,2) -> (2,2) 25: (1,3,1) -> (1,1)
11: (2,1,1) -> (2,1,1) 26: (1,2,2) -> (1,2)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1,1)
13: (1,2,1) -> (1,1) 28: (1,1,3) -> (1,1)
14: (1,1,2) -> (1,1) 29: (1,1,2,1) -> (1,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], Less], {n, 0, 100}]
CROSSREFS
All of the following pertain to compositions in standard order:
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.
+10
39
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], GreaterEqual], {n, 0, 100}]
CROSSREFS
All of the following pertain to compositions in standard order:
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of strictly increasing runs sum to k.
+10
31
1, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 1, 2, 0, 5, 0, 1, 3, 5, 0, 7, 0, 2, 4, 6, 9, 0, 11, 0, 2, 7, 10, 13, 17, 0, 15, 0, 3, 8, 20, 23, 24, 28, 0, 22, 0, 3, 14, 26, 47, 47, 42, 47, 0, 30, 0, 5, 17, 45, 66, 101, 92, 71, 73, 0, 42, 0, 5, 27, 61, 124, 154, 201, 166, 116, 114, 0, 56
COMMENTS
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.
EXAMPLE
Triangle begins:
1
0 1
0 0 2
0 1 0 3
0 1 2 0 5
0 1 3 5 0 7
0 2 4 6 9 0 11
0 2 7 10 13 17 0 15
0 3 8 20 23 24 28 0 22
0 3 14 26 47 47 42 47 0 30
0 5 17 45 66 101 92 71 73 0 42
0 5 27 61 124 154 201 166 116 114 0 56
0 7 33 101 181 300 327 379 291 182 170 0 77
0 8 48 138 307 467 668 656 680 488 282 253 0 101
Row n = 6 counts the following compositions:
. (15) (24) (231) (312) . (6)
(123) (141) (213) (2121) (51)
(114) (132) (2112) (42)
(1212) (1311) (1221) (411)
(1131) (1122) (33)
(1113) (12111) (321)
(11211) (3111)
(11121) (222)
(11112) (2211)
(21111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, Less]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
For length instead of sum we have A333213. Leaders of strictly increasing runs in standard compositions are A374683. The corresponding rank statistic is A374684. Other types of runs (instead of strictly increasing):
- 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 weakly decreasing runs we have A374748. - For leaders of strictly decreasing runs we have A374766. A003242 counts anti-run compositions.
Number of integer compositions of n whose leaders of weakly increasing runs are strictly increasing.
+10
29
1, 1, 2, 3, 5, 7, 12, 17, 28, 43, 67, 103, 162, 245, 374, 569, 854, 1278, 1902, 2816, 4148, 6087, 8881, 12926, 18726, 27042
COMMENTS
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.
EXAMPLE
The composition (1,3,3,2,4,3) has weakly increasing runs ((1,3,3),(2,4),(3)), with leaders (1,2,3), so is counted under a(16).
The a(0) = 1 through a(7) = 17 compositions:
() (1) (2) (3) (4) (5) (6) (7)
(11) (12) (13) (14) (15) (16)
(111) (22) (23) (24) (25)
(112) (113) (33) (34)
(1111) (122) (114) (115)
(1112) (123) (124)
(11111) (132) (133)
(222) (142)
(1113) (223)
(1122) (1114)
(11112) (1123)
(111111) (1132)
(1222)
(11113)
(11122)
(111112)
(1111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Less@@First/@Split[#, LessEqual]&]], {n, 0, 15}]
CROSSREFS
Ranked by positions of strictly increasing rows in A374629 (sums A374630). Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A000041. - For leaders of anti-runs we have A374679. - For leaders of strictly increasing runs we have A374688. - For leaders of strictly decreasing runs we have A374762. Types of run-leaders (instead of strictly increasing):
- 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 weakly increasing leaders we have A374635. A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Number of integer compositions of n whose leaders of weakly increasing runs are identical.
+10
27
1, 1, 2, 3, 6, 10, 19, 34, 63, 116, 218, 405, 763, 1436, 2714, 5127, 9718, 18422, 34968, 66397, 126168, 239820, 456027, 867325, 1649970, 3139288, 5973746, 11368487, 21636909, 41182648, 78389204, 149216039, 284046349, 540722066, 1029362133, 1959609449
COMMENTS
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.
FORMULA
G.f.: 1 + Sum_{i>0} A(x,i) where A(x,i) = 1/(1-x^i) * (x^i + Sum_{z>0} ( ((x^i)/(1-x^i) * (-1 + Product_{j>i} (1/(1-x^j))))^z )) is the g.f. for compositions of this kind with all leaders equal to i. - John Tyler Rascoe, Jul 25 2024
EXAMPLE
The composition (1,3,1,4,1,2,2,1) has maximal weakly increasing subsequences ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so is counted under a(15).
The a(0) = 1 through a(6) = 19 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(121) (122) (114)
(1111) (131) (123)
(1112) (141)
(1121) (222)
(1211) (1113)
(11111) (1122)
(1131)
(1212)
(1221)
(1311)
(11112)
(11121)
(11211)
(12111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], SameQ@@First/@Split[#, LessEqual]&]], {n, 0, 15}]
PROG
(PARI)
C_x(N) = {my(x='x+O('x^N), h=1+sum(i=1, N, 1/(1-x^i)*(x^i+sum(z=1, N-i+1, (x^i/(1-x^i)*(-1+(1/prod(j=i+1, N-i, 1-x^j))))^z)))); Vec(h)}
CROSSREFS
Types of runs (instead of weakly increasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919. - For leaders of strictly increasing runs we have A374686, ranks A374685. - For leaders of weakly decreasing runs we have A374742, ranks A374744. - For leaders of strictly decreasing runs we have A374760, ranks A374759. Types of run-leaders (instead of identical):
- For strictly decreasing leaders we appear to have A188920. - For weakly decreasing leaders we appear to have A189076. - For strictly increasing leaders we have A374634. - For weakly increasing leaders we have A374635. A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A000009, A106356, A124766, A238343, A261982, A333213, A373949, A374518, A374687, A374743, A374761.
Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.
+10
27
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], Greater], {n, 0, 100}]
CROSSREFS
Row-leaders of nonempty rows are A065120. All of the following pertain to compositions in standard order:
- Ranks of non-contiguous compositions are A374253, counted by A335548. Six types of runs:
Cf. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.
Number of integer compositions of n whose leaders of weakly decreasing runs are identical.
+10
24
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
COMMENTS
The weakly decreasing run-leaders of a sequence are obtained by splitting into maximal weakly decreasing subsequences and taking the first term of each.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], SameQ@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
CROSSREFS
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919. - 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 weakly increasing leaders we appear to have A188900. A335456 counts patterns matched by compositions.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374741.
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