Displaying 1-10 of 15 results found.
Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.
+10
35
1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 3, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 4, 4, 1, 3, 3, 3, 3, 3, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1
COMMENTS
Anti-runs summing to n are counted by A003242(n). 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.
EXAMPLE
The maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
The nonnegative integers, corresponding compositions, and leaders of anti-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)
6: (1,2) -> (1) 21: (2,2,1) -> (2,2)
7: (1,1,1) -> (1,1,1) 22: (2,1,2) -> (2)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1,1)
9: (3,1) -> (3) 24: (1,4) -> (1)
10: (2,2) -> (2,2) 25: (1,3,1) -> (1)
11: (2,1,1) -> (2,1) 26: (1,2,2) -> (1,2)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1)
13: (1,2,1) -> (1) 28: (1,1,3) -> (1,1)
14: (1,1,2) -> (1,1) 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], UnsameQ], {n, 0, 100}]
CROSSREFS
Row-leaders of nonempty rows are A065120. 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:
Six types of maximal runs:
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.
Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.
+10
25
0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 32, 33, 34, 35, 37, 38, 40, 41, 44, 48, 49, 50, 52, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 88, 89, 92, 96, 97, 98, 101, 102, 104, 105, 108, 128, 129, 130, 131, 132, 133
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 10000000th composition in standard order is (3,1,4,3,2,1,2,8), with strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1) so 10000000 is in the sequence.
The terms together with the corresponding compositions begin:
0: ()
1: (1)
2: (2)
4: (3)
5: (2,1)
6: (1,2)
8: (4)
9: (3,1)
11: (2,1,1)
12: (1,3)
13: (1,2,1)
16: (5)
17: (4,1)
18: (3,2)
19: (3,1,1)
20: (2,3)
24: (1,4)
25: (1,3,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@First/@Split[stc[#], Greater]&]
CROSSREFS
For identical instead of distinct runs we have A374759, counted by A374760. Compositions of this type are counted by A374761. All of the following pertain to compositions in standard order:
Six types of runs:
Cf. A065120, A106356, A188920, A233564, A238343, A272919, A333213, A373949, A374633, A374685, A374744, A374758, A375128.
Numbers k such that the leaders of anti-runs in the k-th composition in standard order ( A066099) are distinct.
+10
24
0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104
COMMENTS
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.
EXAMPLE
The terms together with corresponding compositions begin:
0: ()
1: (1)
2: (2)
4: (3)
5: (2,1)
6: (1,2)
8: (4)
9: (3,1)
11: (2,1,1)
12: (1,3)
13: (1,2,1)
16: (5)
17: (4,1)
18: (3,2)
19: (3,1,1)
20: (2,3)
22: (2,1,2)
24: (1,4)
25: (1,3,1)
26: (1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@First/@Split[stc[#], UnsameQ]&]
CROSSREFS
Positions of distinct (strict) rows in A374515. Compositions of this type are counted by A374518. For identical instead of distinct we have A374519, counted by A374517. Other types of runs (instead of anti-):
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:
Six types of maximal runs:
Numbers k such that the leaders of anti-runs in the k-th composition in standard order ( A066099) are identical.
+10
23
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 77, 80, 81, 82, 84, 85
COMMENTS
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.
EXAMPLE
The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2), so 346 is in the sequence.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], SameQ@@First/@Split[stc[#], UnsameQ]&]
CROSSREFS
Positions of constant rows in A374515. Compositions of this type are counted by A374517. For distinct instead of identical leaders we have A374638, counted by A374518. Other types of runs (instead of anti-):
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:
Six types of maximal runs:
Weakly increasing run-leader transformation for standard compositions.
+10
22
0, 1, 2, 1, 4, 5, 1, 1, 8, 9, 2, 5, 1, 3, 1, 1, 16, 17, 18, 9, 2, 5, 5, 5, 1, 3, 1, 3, 1, 3, 1, 1, 32, 33, 34, 17, 4, 37, 9, 9, 2, 5, 2, 5, 5, 11, 5, 5, 1, 3, 6, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 64, 65, 66, 33, 68, 69, 17, 17, 4, 9, 18, 37, 9, 19, 9, 9
COMMENTS
The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.
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.
EXAMPLE
The 813th composition in standard order is (1,3,2,1,2,1), with weakly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.
MATHEMATICA
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], LessEqual]], {n, 0, 100}]
CROSSREFS
The strict opposite version is A375126. All of the following pertain to compositions in standard order:
- Run-sum transformation is A353847. Six types of runs:
Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.
+10
21
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
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.
EXAMPLE
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)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], SameQ@@First/@Split[stc[#], Greater]&]
CROSSREFS
For leaders of weakly increasing runs we have A374633, counted by A374631. 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:
- Ranks of non-contiguous compositions are A374253, counted by A335548. Six types of runs:
Cf. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.
Weakly decreasing run-leader transformation for standard compositions.
+10
16
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
The strict opposite version is A375125. All of the following pertain to compositions in standard order:
- Run-sum transformation is A353847. Six types of runs:
Strictly increasing run-leader transformation for standard compositions.
+10
16
0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 1, 3, 3, 15, 16, 17, 18, 19, 2, 21, 5, 23, 1, 3, 6, 7, 3, 7, 7, 31, 32, 33, 34, 35, 36, 37, 9, 39, 2, 5, 42, 43, 5, 11, 11, 47, 1, 3, 6, 7, 1, 13, 3, 15, 3, 7, 14, 15, 7, 15, 15, 63, 64, 65, 66, 67, 68, 69, 17, 71, 4, 73
COMMENTS
The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.
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 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.
MATHEMATICA
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], Less]], {n, 0, 100}]
CROSSREFS
The weak opposite version is A375124. All of the following pertain to compositions in standard order:
- Run-sum transformation is A353847. Six types of runs:
Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374634, A374635, A374684, A374700, A375128.
Strictly decreasing run-leader transformation for standard compositions.
+10
16
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
COMMENTS
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?
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
The weak opposite version is A375123. All of the following pertain to compositions in standard order:
- Run-sum transformation is A353847. Six types of runs:
Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374634, A374685, A374744, A374766, A375128.
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