|
| |
|
|
A206437
|
|
Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1 <= j <= A000041(n).
|
|
99
|
|
|
|
1, 2, 1, 3, 1, 1, 2, 4, 2, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 5, 2, 4, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in lexicographic ordering, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).
Some properties of the regions of n:
- The number of regions of n equals the number of partitions of n (see A000041).
- The set of regions of n contain the sets of regions of all positive integers previous to n.
- The first j regions of n are also first j regions of all integers greater than n.
- The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).
- If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).
- The rank of a region is the largest part minus the number of parts (see A194447).
- The sum of all ranks of the regions of n is equal to zero.
How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j] where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in lexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] up to intercept the next segment in a lower row with respect to the axis "y". So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the Links section.
Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).
Also triangle read by rows in which row r lists the parts of the last section of the set of partitions of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - Omar E. Pol, Apr 28 2012
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
-----------------------------------------
Region Triangle
j of parts
-----------------------------------------
1 1;
2 2,1;
3 3,1,1;
4 2;
5 4,2,1,1,1;
6 3;
7 5,2,1,1,1,1,1;
8 2;
9 4,2;
10 3;
11 6,3,2,2,1,1,1,1,1,1,1;
12 3;
13 5,2;
14 4;
15 7,3,2,2,1,1,1,1,1,1,1,1,1,1,1;
.
The rotated triangle shows each row as a partition:
.
. 7
. 4 3
. 5 2
. 3 2 2
. 6 1
. 3 3 1
. 4 2 1
. 2 2 2 1
. 5 1 1
. 3 2 1 1
. 4 1 1 1
. 2 2 1 1 1
. 3 1 1 1 1
. 2 1 1 1 1 1
1 1 1 1 1 1 1
.
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the parts of the last section of the set of partitions of r ordered by regions (see comments):
[1];
[2,1];
[3,1,1];
[2],[4,2,1,1,1];
[3],[5,2,1,1,1,1,1];
[2],[4,2],[3],[6,3,2,2,1,1,1,1,1,1,1];
[3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1];
|
|
|
MATHEMATICA
|
lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
reg = {}; l = {};
For[j = 1, j <= 22, j++,
mx = Max@lex[j][[j]]; AppendTo[l, mx];
For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
AppendTo[reg, Take[Reverse[First /@ lex[mx]], j - i]];
];
Flatten@reg (* Robert Price, Apr 21 2020, revised Jul 24 2020 *)
|
|
|
CROSSREFS
|
Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A194436-A194439, A194447, A194448, A196025, A198381.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Further edited by Omar E. Pol, Mar 31 2012, Jan 27 2013
|
|
|
STATUS
|
approved
|
| |
|
|
