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A355522
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Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.
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0
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2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 6, 6, 2, 2, 1, 1, 3, 10, 6, 5, 2, 2, 1, 1, 4, 11, 11, 6, 4, 2, 2, 1, 1, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1
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OFFSET
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2,1
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COMMENTS
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The triangle starts with n = 2, and k ranges from 0 to n - 2.
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LINKS
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EXAMPLE
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Triangle begins:
2
2 1
3 1 1
2 3 1 1
4 3 2 1 1
2 6 3 2 1 1
4 6 6 2 2 1 1
3 10 6 5 2 2 1 1
4 11 11 6 4 2 2 1 1
2 16 13 10 5 4 2 2 1 1
6 17 19 12 9 4 4 2 2 1 1
2 24 24 18 11 8 4 4 2 2 1 1
4 27 34 22 17 10 7 4 4 2 2 1 1
4 35 39 33 20 15 9 7 4 4 2 2 1 1
5 39 56 39 30 19 14 8 7 4 4 2 2 1 1
For example, row n = 8 counts the following reversed partitions:
(8) (233) (35) (125) (26) (116) (17)
(44) (1223) (134) (11114) (1115)
(2222) (11123) (224)
(11111111) (11222) (1124)
(111122) (1133)
(1111112) (111113)
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MATHEMATICA
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Table[Length[Select[Reverse/@IntegerPartitions[n], If[Length[#]==1, 0, Max@@Differences[#]]==k&]], {n, 2, 15}, {k, 0, n-2}]
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CROSSREFS
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Crossrefs found in the link are not repeated here.
This is a trimmed version of A238353, which extends to k = n.
For minimum instead of maximum we have A238354.
Ignoring singletons entirely gives A238710.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A279945 counts partitions by number of distinct differences.
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KEYWORD
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AUTHOR
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STATUS
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approved
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