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A095684
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Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.
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11
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1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 1, 1, 2, 2, 3, 1, 1, 2, 3, 3, 1, 1, 2, 3, 4, 1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3
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OFFSET
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1,5
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COMMENTS
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Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition 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. For example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - Gus Wiseman, Apr 26 2020
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LINKS
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EXAMPLE
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1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, ...
The 8 strings of length 4 are 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234.
The triangle read by columns begins:
1:{1} 2:{1,1} 4:{1,1,1} 8:{1,1,1,1} 16:{1,1,1,1,1}
3:{1,2} 5:{1,1,2} 9:{1,1,1,2} 17:{1,1,1,1,2}
6:{1,2,2} 10:{1,1,2,2} 18:{1,1,1,2,2}
7:{1,2,3} 11:{1,1,2,3} 19:{1,1,1,2,3}
12:{1,2,2,2} 20:{1,1,2,2,2}
13:{1,2,2,3} 21:{1,1,2,2,3}
14:{1,2,3,3} 22:{1,1,2,3,3}
15:{1,2,3,4} 23:{1,1,2,3,4}
24:{1,2,2,2,2}
25:{1,2,2,2,3}
26:{1,2,2,3,3}
27:{1,2,2,3,4}
28:{1,2,3,3,3}
29:{1,2,3,3,4}
30:{1,2,3,4,4}
31:{1,2,3,4,5}
(End)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i, y[[i]]], {i, Length[y]}];
Table[ptnToNorm[stc[n]], {n, 15}] (* Gus Wiseman, Apr 26 2020 *)
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CROSSREFS
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The number of distinct parts in row n is A000120(n), also the maximum part.
The version for prime indices is A305936.
There are A333942(n) multiset partitions of row n.
Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are A269134.
All of the following pertain to compositions in standard order (A066099):
- Constant compositions are A272919.
- Distinct parts are counted by A334028.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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