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A279945
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Irregular triangular array: t(n,k) = number of partitions of n having lexicographic difference set of size k; see Comments.
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18
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1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 6, 4, 1, 4, 10, 1, 6, 14, 1, 1, 8, 17, 4, 1, 8, 27, 6, 1, 6, 36, 13, 1, 13, 42, 21, 1, 7, 58, 35, 1, 10, 72, 52, 1, 15, 75, 84, 1, 1, 12, 106, 107, 5, 1, 9, 119, 159, 9, 1, 19, 142, 204, 19, 1, 10, 164, 283, 32, 1, 16, 199
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OFFSET
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1,5
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COMMENTS
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A partition P = [p(1), p(2), ..., p(k)] with p(1) >= p(2) >= ... >= p(k) has lexicographic difference set {0} union {|p(i) - p(i-1)|: 2 <= i <= k}. Column 2 is A049990, and the n-th row sum is A000041(n).
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LINKS
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EXAMPLE
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First 20 rows of array:
1
1 1
1 2
1 3 1
1 3 3
1 6 4
1 4 10
1 6 14 1
1 8 17 4
1 8 27 6
1 6 36 13
1 13 42 21
1 7 58 35
1 10 72 52
1 15 75 84 1
1 12 106 107 5
1 9 119 159 9
1 19 142 204 19
1 10 164 283 32
1 16 199 360 51
Row 5: the 7 partitions of 5 are shown here with difference sets:
partition difference set size
[5] null 0
[4,1] {3} 1
[3,2] {1} 1
[3,1,1] {0,2} 2
[2,2,1] {0,1} 2
[2,1,1,1] {0,1} 2
[1,1,1,1] {0} 1
Row 5 of the array is 1 3 3, these being the number of 0's, 1's, 2's in the "size" column.
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MATHEMATICA
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p[n_] := IntegerPartitions[n]; z = 20;
t[n_, k_] := Length[DeleteDuplicates[Abs[Differences[p[n][[k]]]]]];
u[n_] := Table[t[n, k], {k, 1, PartitionsP[n]}];
v = Table[Count[u[n], h], {n, 1, z}, {h, 0, Max[u[n]]}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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