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%I #49 Apr 26 2020 21:52:18
%S 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 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 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 Triangle of regions and partitions of integers (see Comments lines for definition).
%C 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 Robert Price, <a href="/A193870/b193870.txt">Table of n, a(n) for n = 1..196878, rows 1-627.</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F T(n,1) = A141285(n).
%F T(n,k) = A167392(n), if k = n.
%e Triangle begins:
%e 1,
%e 2, 1,
%e 3, 1, 1,
%e 2, 0, 0, 0,
%e 4, 2, 1, 1, 1,
%e 3, 0, 0, 0, 0, 0,
%e 5, 2, 1, 1, 1, 1, 1,
%e 2, 0, 0, 0, 0, 0, 0, 0,
%e 4, 2, 0, 0, 0, 0, 0, 0, 0,
%e 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1,
%e …
%e 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 A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
%t "Table"], {_, _}][[All, 2]];
%t A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
%t "Table"], {_, _}][[All, 2]];f[n_] := Module[{v},
%t v = Take[A206437, A194446[[n]]];
%t A206437 = Drop[A206437, A194446[[n]]];
%t PadRight[v, n]];
%t Table[f[n], {n, PartitionsP[20]}] // Flatten (* _Robert Price_, Apr 26 2020 *)
%Y Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392.
%Y Cf. A046746, A135010, A138121, A182699, A182709, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Aug 07 2011
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