A279945 - OEIS

%I #16 Dec 27 2016 23:25:47

%S 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 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 119,159,9,1,19,142,204,19,1,10,164,283,32,1,16,199

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

%C 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 Clark Kimberling, <a href="/A279945/b279945.txt">Table of n, a(n) for n = 1..100</a>

%e First 20 rows of array:

%e 1

%e 1    1

%e 1    2

%e 1    3    1

%e 1    3    3

%e 1    6    4

%e 1    4    10

%e 1    6    14    1

%e 1    8    17    4

%e 1    8    27    6

%e 1    6    36    13

%e 1    13   42    21

%e 1    7    58    35

%e 1    10   72    52

%e 1    15   75    84    1

%e 1    12   106   107   5

%e 1    9    119   159   9

%e 1    19   142   204   19

%e 1    10   164   283   32

%e 1    16   199   360   51

%e Row 5: the 7 partitions of 5 are shown here with difference sets:

%e partition  difference set     size

%e [5]          null              0

%e [4,1]        {3}               1

%e [3,2]        {1}               1

%e [3,1,1]      {0,2}             2

%e [2,2,1]      {0,1}             2

%e [2,1,1,1]    {0,1}             2

%e [1,1,1,1]    {0}               1

%e 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 p[n_] := IntegerPartitions[n]; z = 20;

%t t[n_, k_] := Length[DeleteDuplicates[Abs[Differences[p[n][[k]]]]]];

%t u[n_] := Table[t[n, k], {k, 1, PartitionsP[n]}];

%t v = Table[Count[u[n], h], {n, 1, z}, {h, 0, Max[u[n]]}]

%t TableForm[v] (* A279945 array *)

%t Flatten[v]   (* A279945 sequence *)

%Y Cf. A000041, A049990.

%K nonn,easy

%O 1,5

%A _Clark Kimberling_, Dec 26 2016