A193870 - OEIS

A193870

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

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

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.

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:

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.

MATHEMATICA

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",

"Table"], {_, _}][[All, 2]];

A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",

"Table"], {_, _}][[All, 2]]; f[n_] := Module[{v},

v = Take[A206437, A194446[[n]]];

A206437 = Drop[A206437, A194446[[n]]];

PadRight[v, n]];

Table[f[n], {n, PartitionsP[20]}] // Flatten (* Robert Price, Apr 26 2020 *)

CROSSREFS

Mirror of triangle A186114. Column 1 gives A141285. Right diagonal gives A167392.