A211009 Triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j), 1<=k<=A141285(n).

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 11, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 15, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 22

Offset

1,3

Comments

Also the finite sequence a(1)..a(r), where a(r) is a record in the sequence, is also a finite triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of the integer whose number of partitions is equal to a(r).

T(n,k) is also 1 plus the number of holes between T(n,k) and the previous member in the column k of triangle.

T(n,k) is also the height of the column mentioned in the definition, in a three-dimensional model of the set of partitions of j, in which the regions appear rotated 90 degrees and where the pivots are the largest part of every region (see A141285). For the definition of "region" see A206437. - Omar E. Pol, Feb 06 2014

Links

Table of n, a(n) for n=1..86.

Omar E. Pol, Illustration of the seven regions of 5

Example

The irregular triangle begins:
1;
1, 2;
1, 1, 3;
1, 1;
1, 1, 2, 5;
1, 1, 1;
1, 1, 1, 2, 7;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 11;
1, 1, 1;
1, 1, 1, 2, 2;
1, 1, 1, 1;
1, 1, 1, 1, 2, 4, 15;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 4;
1, 1, 1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 2, 3, 7, 22;
...
From Omar E. Pol, Feb 06 2014: (Start)
Illustration of initial terms:
.    _
.   |_|
.    1
.      _
.    _|_|
.   |_ _|
.    1 2
.        _
.       |_|
.    _ _|_|
.   |_ _ _|
.    1 1 3
.    _ _
.   |_ _|
.    1 1
.          _
.         |_|
.         |_|
.        _|_|
.    _ _|_ _|
.   |_ _ _ _|
.    1 1 2 5
.
(End)

Crossrefs

Records give positive terms of A000041. Row n has length A141285(n). Row sums give A186412.

Cf. A040051, A135010, A182244, A182703, A186114, A187219, A193870, A194446, A206437.

Sequence in context: A331368 A106177 A211985 * A337266 A211986 A211989

Adjacent sequences: A211006 A211007 A211008 * A211010 A211011 A211012

Keyword

nonn,tabf

Author

Omar E. Pol, Oct 21 2012

Extensions

Better definition from Omar E. Pol, Feb 06 2014

Status

approved