Revision History for A353932
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#6 by Michael De Vlieger at Sun Jun 12 22:52:25 EDT 2022
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#5 by Gus Wiseman at Sun Jun 12 21:28:24 EDT 2022
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#4 by Gus Wiseman at Sun Jun 12 16:00:41 EDT 2022
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| CROSSREFS
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`A353863 counts run-sum-complete partitions.
Cf. A003242. , A175413, A181819, A238279, A274174, A333381, A333489, A333755, `, A353835, A353839, `, A353863, A353864, `A353866.
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#3 by Gus Wiseman at Sat Jun 11 00:29:11 EDT 2022
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| EXAMPLE
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For example, composition 350 in standard order is (2,2,1,1,1,2), andso row its350 run-sumsis (4,3,2) are listed in row 350.).
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#2 by Gus Wiseman at Fri Jun 10 13:25:15 EDT 2022
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| NAME
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allocatedIrregular triangle read by rows where row k lists the run-sums of the k-th composition forin Gusstandard Wisemanorder.
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| DATA
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1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 3, 1, 4, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 5, 4, 1, 3, 2, 3, 2, 2, 3, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 1, 1, 4, 1, 2, 2, 2, 3, 2, 2, 1, 3, 2, 5, 6, 5, 1, 4, 2, 4, 2, 6, 3, 2, 1, 3, 1, 2, 3, 3, 2, 4, 2, 3, 1, 6, 4, 2, 2, 1, 3
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| OFFSET
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1,2
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| COMMENTS
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Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
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.
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| EXAMPLE
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Triangle begins:
1
2
2
3
2 1
1 2
3
4
3 1
4
2 2
1 3
1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), and its run-sums (4,3,2) are listed in row 350.
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| MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total/@Split[stc[n]], {n, 0, 30}]
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| CROSSREFS
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Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
`A353863 counts run-sum-complete partitions.
Cf. A003242. A175413, A181819, A238279, A274174, A333381, A333489, A333755, `A353835, A353839, `A353864, `A353866.
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| KEYWORD
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allocated
nonn,tabf
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| AUTHOR
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Gus Wiseman, Jun 10 2022
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Wed May 11 14:47:45 EDT 2022
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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