A186114 - OEIS

A186114

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

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Let r = T(n,k) be a record in the sequence. The consecutive records "r" are the natural numbers A000027. Consider the first n rows; the triangle T(n,k) has the property that the columns, without the zeros, from k..1, are also the partitions of r in juxtaposed reverse-lexicographical order, so k is also A000041(r), the number of partitions of r. Note that a record r is always the final term of a row if such row contains 1’s. The number of positive integer a(1)..r is A006128(r). The sums a(1)..r is A066186(r). Here the set of positive integers in every row (from 1 to n) is called a “region” of r. The number of regions of r equals the number of partitions of r. If T(n,1) = 1 then the row n is formed by the smallest parts, in nondecreasing order, of all partitions of T(n,n).

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) = A167392(n).

T(n,k) = A141285(n), if k = n.

EXAMPLE

Triangle begins:

1, 2,

1, 1, 3,

0, 0, 0, 2,

1, 1, 1, 2, 4,

0, 0, 0, 0, 0, 3,

1, 1, 1, 1, 1, 2, 5,

0, 0, 0, 0, 0, 0, 0, 2,

0, 0, 0, 0, 0, 0, 0, 2, 4,

0, 0, 0, 0, 0, 0, 0, 0, 0, 3,

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6

...

The row n = 11 contains the 6th record in the sequence: a(66) = T(11,11) = 6, then consider the first 11 rows of triangle. Note that the columns, from k = 11..1, without the zeros, are also the 11 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]]];

Reverse[PadRight[v, n]]];

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

CROSSREFS

Mirror of triangle A193870. Column 1 gives A167392. Right diagonal gives A141285.