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A115622
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Irregular triangle read by rows: row m lists the signatures of all partitions of m when the partitions are arranged in Mathematica order.
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6
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1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 2, 4, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 4, 1, 3, 1, 3, 2, 5, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 2, 1, 2, 2, 2
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OFFSET
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1,3
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COMMENTS
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The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g., [4,4,3,1,1] = [4^2,3^1,1^2], so the repetition factors are 2,1,2, making the signature [2,2,1] = [2^2,1].
The sum (or order) of the signature is the number of parts of the original partition and the number of parts of the signature is the number of distinct parts of the original partition.
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LINKS
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EXAMPLE
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The first six rows of the triangle are as follows.
1: [1]
2: [1] [2]
3: [1] [1,1] [3]
4: [1] [1,1] [2] [2,1] [4]
5: [1] [1,1] [1,1] [2,1] [2,1] [3,1] [5]
6: [1] [1,1] [1,1] [2,1] [2] [1,1,1] [3,1] [3] [2,2] [4,1] [6]
See A115621 for the signatures in Abramowitz-Stegun order.
(End)
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MATHEMATICA
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(* row[] and triangle[] compute structured rows of the triangle as laid out above *)
mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]
row[n_] := Map[Reverse[Sort[#]]&, mL[IntegerPartitions[n]]]
triangle[n_] := Map[row, Range[n]]
a115622[n_]:= Flatten[triangle[n]]
Map[Sort[#, Greater] &, Table[Last /@ Transpose /@ Tally /@ IntegerPartitions[n], {n, 8}], 2] // Flatten (* Robert Price, Jun 12 2020 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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