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A359902
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Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.
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65
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1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 1, 8, 6, 3, 2, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 1, 14, 11, 5, 4, 2, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,11
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COMMENTS
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The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
1 0 1
1 0 0 1
2 1 0 0 1
2 2 0 0 0 1
4 2 1 0 0 0 1
4 3 2 0 0 0 0 1
7 4 3 1 0 0 0 0 1
8 6 3 2 0 0 0 0 0 1
12 8 4 3 1 0 0 0 0 0 1
14 11 5 4 2 0 0 0 0 0 0 1
21 14 8 4 3 1 0 0 0 0 0 0 1
24 20 10 5 4 2 0 0 0 0 0 0 0 1
34 25 15 6 5 3 1 0 0 0 0 0 0 0 1
For example, row n = 9 counts the following partitions:
(7,1,1) (5,2,2) (3,3,3) (4,4,1) . . . . (9)
(3,3,1,1,1) (6,2,1) (4,3,2)
(4,2,1,1,1) (2,2,2,2,1) (5,3,1)
(5,1,1,1,1) (3,2,2,1,1)
(2,2,1,1,1,1,1)
(3,1,1,1,1,1,1)
(1,1,1,1,1,1,1,1,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Median[#]==k&]], {n, 15}, {k, n}]
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CROSSREFS
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The median statistic is ranked by A360005(n)/2.
A240219 counts partitions w/ the same mean as median, complement A359894.
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KEYWORD
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AUTHOR
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STATUS
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approved
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