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A372120
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Numbers k such that the k-th composition in standard order is biquanimous.
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3
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0, 3, 10, 11, 13, 14, 15, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 136, 137, 138, 139, 140, 141, 142, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 165, 166, 167, 168, 169
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OFFSET
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1,2
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COMMENTS
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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.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
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LINKS
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EXAMPLE
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The terms and corresponding compositions begin:
0: ()
3: (1,1)
10: (2,2)
11: (2,1,1)
13: (1,2,1)
14: (1,1,2)
15: (1,1,1,1)
36: (3,3)
37: (3,2,1)
38: (3,1,2)
39: (3,1,1,1)
41: (2,3,1)
43: (2,2,1,1)
44: (2,1,3)
45: (2,1,2,1)
46: (2,1,1,2)
47: (2,1,1,1,1)
50: (1,3,2)
51: (1,3,1,1)
52: (1,2,3)
53: (1,2,2,1)
54: (1,2,1,2)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]
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CROSSREFS
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These compositions are counted by A064914.
The unordered version (integer partitions) is A357976, counted by A002219.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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