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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.

%I A279945 #16 Dec 27 2016 23:25:47
%S A279945 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,
%T A279945 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,
%U A279945 119,159,9,1,19,142,204,19,1,10,164,283,32,1,16,199
%N A279945 Irregular triangular array: t(n,k) = number of partitions of n having lexicographic difference set of size k; see Comments.
%C A279945 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).
%H A279945 Clark Kimberling, <a href="/A279945/b279945.txt">Table of n, a(n) for n = 1..100</a>
%e A279945 First 20 rows of array:
%e A279945 1
%e A279945 1    1
%e A279945 1    2
%e A279945 1    3    1
%e A279945 1    3    3
%e A279945 1    6    4
%e A279945 1    4    10
%e A279945 1    6    14    1
%e A279945 1    8    17    4
%e A279945 1    8    27    6
%e A279945 1    6    36    13
%e A279945 1    13   42    21
%e A279945 1    7    58    35
%e A279945 1    10   72    52
%e A279945 1    15   75    84    1
%e A279945 1    12   106   107   5
%e A279945 1    9    119   159   9
%e A279945 1    19   142   204   19
%e A279945 1    10   164   283   32
%e A279945 1    16   199   360   51
%e A279945 Row 5: the 7 partitions of 5 are shown here with difference sets:
%e A279945 partition  difference set     size
%e A279945 [5]          null              0
%e A279945 [4,1]        {3}               1
%e A279945 [3,2]        {1}               1
%e A279945 [3,1,1]      {0,2}             2
%e A279945 [2,2,1]      {0,1}             2
%e A279945 [2,1,1,1]    {0,1}             2
%e A279945 [1,1,1,1]    {0}               1
%e A279945 Row 5 of the array is 1 3 3, these being the number of 0's, 1's, 2's in the "size" column.
%t A279945 p[n_] := IntegerPartitions[n]; z = 20;
%t A279945 t[n_, k_] := Length[DeleteDuplicates[Abs[Differences[p[n][[k]]]]]];
%t A279945 u[n_] := Table[t[n, k], {k, 1, PartitionsP[n]}];
%t A279945 v = Table[Count[u[n], h], {n, 1, z}, {h, 0, Max[u[n]]}]
%t A279945 TableForm[v] (* A279945 array *)
%t A279945 Flatten[v]   (* A279945 sequence *)
%Y A279945 Cf. A000041, A049990.
%K A279945 nonn,easy
%O A279945 1,5
%A A279945 _Clark Kimberling_, Dec 26 2016