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A359893
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Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.
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132
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1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,10
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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
1 0 1
1 1 0 0 1
2 0 2 0 0 0 1
3 0 1 2 0 0 0 0 1
4 1 2 0 3 0 0 0 0 0 1
6 1 3 0 1 3 0 0 0 0 0 0 1
8 1 6 0 2 0 4 0 0 0 0 0 0 0 1
11 2 7 1 3 0 1 4 0 0 0 0 0 0 0 0 1
15 2 10 3 4 0 2 0 5 0 0 0 0 0 0 0 0 0 1
20 3 13 3 7 0 3 0 1 5 0 0 0 0 0 0 0 0 0 0 1
26 4 19 3 11 1 4 0 2 0 6 0 0 0 0 0 0 0 0 0 0 0 1
For example, row n = 8 counts the following partitions:
611 4211 422 . 332 . 44 . . . . . . . 8
5111 521 431 53
32111 2222 62
41111 3221 71
221111 3311
311111 22211
2111111
11111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Median[#]==k&]], {n, 1, 10}, {k, 1, n, 1/2}]
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CROSSREFS
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Row lengths are 2n-1 = A005408(n-1).
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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nonn,tabf
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AUTHOR
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STATUS
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approved
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