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A237513
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T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.
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1
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 10, 1, 5, 1, 2, 1, 1, 1, 13, 1, 7, 1, 3, 2, 1, 1, 1, 17, 1, 10, 1, 5, 3, 2, 1, 1, 1, 21, 1, 13, 1, 7, 6, 3, 1, 1, 1, 1, 26, 1, 17, 1, 10, 10, 5, 1, 3, 1, 1, 1, 31, 1
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OFFSET
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1,10
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COMMENTS
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Column n consists solely of 1's if and only if n = 1 or n is a prime. Column 4: A033638.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 ... 1
1 ... 1 ... 1
1 ... 1 ... 1 ... 3
1 ... 1 ... 1 ... 3 ... 1
1 ... 1 ... 1 ... 5 ... 1 ... 2
1 ... 1 ... 1 ... 7 ... 1 ... 3 ... 1
1 ... 1 ... 1 .. 10 ... 1 ... 5 ... 1 ... 2
1 ... 1 ... 1 .. 13 ... 1 ... 7 ... 1 ... 3 ... 2
1 ... 1 ... 1 .. 17 ... 1 .. 10 ... 1 ... 5 ... 3 ... 2
1 ... 1 ... 1 .. 21 ... 1 .. 13 ... 1 ... 7 ... 6 ... 3 ... 1
1 ... 1 ... 1 .. 26 ... 1 .. 17 ... 1 .. 10 .. 10 ... 5 ... 1 ... 3
The dimensions of maximal horizontal rectangles for the partitions of 6 are 1 X 6, 1 X 5, 2 X 2, 1 X 4, 2 X 3, 2 X 2, 1 X 3, 2 X 2, 2 X 2, 1 X 2, 1 X 1; the numbers of nodes are the products 6, 5, 4, 4, 6, 4, 3, 4, 4, 2, 1; counting the occurrences of 1,2,3,4,5,6 gives 1,1,1,5,1,2, which is row 6 of the array.
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MATHEMATICA
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durfeeSquare[part_] := Max[Map[Min, Transpose[{#, Range[Length[#]]}] &[part]]]; hRectangle[part_] := {#, part[[#]]} &[durfeeSquare[part]]; Map[Last[Transpose[Tally[Sort[Map[Times @@ hRectangle[#] &, IntegerPartitions[#]]]]]] &, Range[20]] (* Peter J. C. Moses, Feb 10 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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