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A279945
Irregular triangular array: t(n,k) = number of partitions of n having lexicographic difference set of size k; see Comments.
18
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
OFFSET
1,5
COMMENTS
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).
LINKS
EXAMPLE
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.
MATHEMATICA
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]]}]
TableForm[v] (* A279945 array *)
Flatten[v] (* A279945 sequence *)
CROSSREFS
Sequence in context: A088742 A256435 A367412 * A342724 A347046 A300322
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 26 2016
STATUS
approved