A279945 Irregular triangular array: t(n,k) = number of partitions of n having lexicographic difference set of size k; see Comments.

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

Author

Clark Kimberling, Dec 26 2016

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000041, A049990.

Programs

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 *)