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A352517
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Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.
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15
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0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2
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OFFSET
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0,7
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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. See also A000120, A059893, A070939, A114994, A225620.
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LINKS
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EXAMPLE
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The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]], #<=y[[#]]&]];
Table[pdw[stc[n]], {n, 0, 30}]
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CROSSREFS
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Positive positions of first appearances are A164894.
The version for partitions is A257990.
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
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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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