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Irregular triangular array, read by rows: row n gives the numbers max(p) - min(p), in nondecreasing order, where p ranges through the strict partitions p of n.
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%I #6 Nov 12 2023 22:00:36

%S 0,0,0,1,0,2,0,1,3,0,2,2,4,0,1,3,3,5,0,2,3,4,4,6,0,1,2,3,4,5,5,7,0,2,

%T 3,3,4,4,5,6,6,8,0,1,3,3,4,4,5,5,6,7,7,9,0,2,2,4,4,4,5,5,5,6,6,7,8,8,

%U 10,0,1,3,3,4,4,5,5,5,6,6,6,7,7,8,9,9

%N Irregular triangular array, read by rows: row n gives the numbers max(p) - min(p), in nondecreasing order, where p ranges through the strict partitions p of n.

%e First twelve rows:

%e 0

%e 0

%e 0 1

%e 0 2

%e 0 1 3

%e 0 2 2 4

%e 0 1 3 3 5

%e 0 2 3 4 4 6

%e 0 1 2 3 4 5 5 7

%e 0 2 3 3 4 4 5 6 6 8

%e 0 1 3 3 4 4 5 5 6 7 7 9

%e 0 2 2 4 4 4 5 5 5 6 6 7 8 8 10

%e The strict partitions p of 8 are 8, 71, 62, 53, 521, 431, for which max(p)-min(p) are the numbers 0, 6, 4, 2, 4, 3, or, in nondecreasing order, 0, 2, 3, 4, 4, 6, as in row 8.

%t z = 20; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &];

%t p[n_, k_] := p[n, k] = d[n][[k]];

%t t = Table[Max[p[n, k]] - Min[p[n, k]], {n, 1, z}, {k, 1, PartitionsQ[n]}]

%t m = Map[Sort, t]

%t Column[m]

%t Flatten[m]

%Y Cf. A000009 (strict partitions), A117454 (frequencies).

%K nonn,tabf

%O 1,6

%A _Clark Kimberling_, Oct 25 2023