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 A193870 Triangle of regions and partitions of integers (see Comments lines for definition). 40
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Omar E. Pol, Illustration of the seven regions of 5 FORMULA T(n,1) = A141285(n). T(n,k) = A167392(n), if k = n. EXAMPLE Triangle begins: 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, 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, … 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. CROSSREFS Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392. Cf. A046746, A135010, A138121, A182699, A182709, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437. Sequence in context: A300977 A144869 A247564 * A058564 A226006 A210943 Adjacent sequences:  A193867 A193868 A193869 * A193871 A193872 A193873 KEYWORD nonn,tabl AUTHOR Omar E. Pol, Aug 07 2011 STATUS approved

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Last modified May 24 15:25 EDT 2019. Contains 323532 sequences. (Running on oeis4.)