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

1,5

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

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

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

Cf. A000041, A049990.

Sequence in context: A088742 A256435 A367412 * A342724 A347046 A300322

Adjacent sequences: A279942 A279943 A279944 * A279946 A279947 A279948

Keyword

nonn,easy

Author

Clark Kimberling, Dec 26 2016

Status

approved