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A193870 Triangle of regions and partitions of integers (see Comments lines for definition).

%I A193870 #49 Apr 26 2020 21:52:18
%S A193870 1,2,1,3,1,1,2,0,0,0,4,2,1,1,1,3,0,0,0,0,0,5,2,1,1,1,1,1,2,0,0,0,0,0,
%T A193870 0,0,4,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,6,3,2,2,1,1,1,1,1,1,1,3,0,
%U A193870 0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,7,3,2,2,1,1,1,1,1,1,1,1,1,1,1
%N A193870 Triangle of regions and partitions of integers (see Comments lines for definition).
%C A193870 Triangle T(n,k) read by rows in which, from rows 1..n, if r = T(n,k) is a record in the sequence then the set of positive integers in every row (from 1 to n) is called a “region” of r. Note that n, the number of regions of r is also the number of partitions of r. The consecutive records "r" are the natural numbers A000027. The triangle has the property that, for rows n..1, the diagonals (without the zeros) are also the partitions of r, in juxtaposed reverse-lexicographical order. Note that a record "r" is the initial term of a row if such row contains 1’s. If T(n,k) is a record in the sequence then A000041(T(n,k)) = n. Note that if T(n,k) < 2 is not the last term of the row n then T(n,k+1) = T(n,k). The union of the rows that contain 1's gives A182715.
%H A193870 Robert Price, <a href="/A193870/b193870.txt">Table of n, a(n) for n = 1..196878, rows 1-627.</a>
%H A193870 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F A193870 T(n,1) = A141285(n).
%F A193870 T(n,k) = A167392(n), if k = n.
%e A193870 Triangle begins:
%e A193870 1,
%e A193870 2, 1,
%e A193870 3, 1, 1,
%e A193870 2, 0, 0, 0,
%e A193870 4, 2, 1, 1, 1,
%e A193870 3, 0, 0, 0, 0, 0,
%e A193870 5, 2, 1, 1, 1, 1, 1,
%e A193870 2, 0, 0, 0, 0, 0, 0, 0,
%e A193870 4, 2, 0, 0, 0, 0, 0, 0, 0,
%e A193870 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e A193870 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1,
%e A193870 …
%e A193870 For n = 11 note that the row n contains the 6th record in the sequence: T(11,1) = a(56) = 6, then consider the first 11 rows of triangle. Note that the diagonals d, from d = n..1, without the zeros, are also the partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. See A026792.
%t A193870 A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
%t A193870      "Table"], {_, _}][[All, 2]];
%t A193870 A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
%t A193870      "Table"], {_, _}][[All, 2]];f[n_] := Module[{v},
%t A193870    v = Take[A206437, A194446[[n]]];
%t A193870    A206437 = Drop[A206437, A194446[[n]]];
%t A193870    PadRight[v, n]];
%t A193870 Table[f[n], {n, PartitionsP[20]}] // Flatten (* _Robert Price_, Apr 26 2020 *)
%Y A193870 Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392.
%Y A193870 Cf. A046746, A135010, A138121, A182699, A182709, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.
%K A193870 nonn,tabl
%O A193870 1,2
%A A193870 _Omar E. Pol_, Aug 07 2011