cp's OEIS Frontend

A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

%I A211989 #15 Jul 18 2022 22:47:44
%S A211989 1,2,1,1,3,1,1,1,2,1,4,2,2,2,1,1,1,1,1,1,3,1,5,3,2,3,1,1,1,1,1,1,1,2,
%T A211989 1,1,1,2,2,1,4,1,6,3,3,4,2,2,2,2,4,1,1,2,2,1,1,2,1,1,1,1,1,1,1,1,1,1,
%U A211989 3,1,1,1,3,2,1,5,1,7,4,3,5,2,3,2,2,5,1,1,3,2,1,1,3,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,4,1,1,1,2,2,2,1,4,2,1,3,3,1,6,1
%N A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.
%C A211989 The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
%C A211989 At stage 1, we start with 1.
%C A211989 At stage j > 1, we write the partitions of j using the following rules:
%C A211989 First, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, starting with the partition that contains the part of size j.
%C A211989 Second, we copy from this array the partitions of j-1 in descending order, as a mirror image, starting with the partition that contains the part of size j-2 together with the part of size 1. At the end of each new row, we added a part of size 1.
%e A211989 A table of partitions.
%e A211989 --------------------------------------------
%e A211989 .              Expanded       Geometric
%e A211989 Partitions     arrangement    model
%e A211989 --------------------------------------------
%e A211989 1;             1;             |*|
%e A211989 --------------------------------------------
%e A211989 2;             . 2;           |* *|
%e A211989 1,1;           1,1;           |o|*|
%e A211989 --------------------------------------------
%e A211989 3;             . . 3;         |* * *|
%e A211989 1,1,1;         1,1,1;         |o|o|*|
%e A211989 2,1;           . 2,1;         |o o|*|
%e A211989 --------------------------------------------
%e A211989 4;             . . . 4;       |* * * *|
%e A211989 2,2;           . 2,. 2;       |* *|* *|
%e A211989 2,1,1;         . 2,1,1;       |o o|o|*|
%e A211989 1,1,1,1;       1,1,1,1;       |o|o|o|*|
%e A211989 3,1;           . . 3,1;       |o o o|*|
%e A211989 --------------------------------------------
%e A211989 5;             . . . . 5;     |* * * * *|
%e A211989 3,2;           . . 3,. 2;     |* * *|* *|
%e A211989 3,1,1;         . . 3,1,1;     |o o o|o|*|
%e A211989 1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
%e A211989 2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
%e A211989 2,2,1;         . 2,. 2,1;     |o o|o o|*|
%e A211989 4,1;           . . . 4,1;     |o o o o|*|
%e A211989 --------------------------------------------
%e A211989 6;             . . . . . 6;   |* * * * * *|
%e A211989 3,3;           . . 3,. . 3;   |* * *|* * *|
%e A211989 4,2;           . . . 4,. 2;   |* * * *|* *|
%e A211989 2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
%e A211989 4,1,1;         . . . 4,1,1;   |o o o o|o|*|
%e A211989 2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
%e A211989 2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
%e A211989 1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
%e A211989 3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
%e A211989 3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
%e A211989 5,1;           . . . . 5,1;   |o o o o o|*|
%e A211989 --------------------------------------------
%e A211989 Note that * is a unitary element of every part of the last section of j.
%Y A211989 Rows sums give A036042, n>=1.
%Y A211989 Other versions are A211983, A211984, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
%Y A211989 Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.
%K A211989 nonn,tabf
%O A211989 1,2
%A A211989 _Omar E. Pol_, Aug 18 2012