A196025 -id:A196025 - cp's OEIS frontend

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).

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

Offset: 1

Omar E. Pol, Feb 14 2012

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 colexicographic order, 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 colexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] down to intercept the next segment in a lower row. 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

			-------------------------------------------
  Region j   Triangle 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];

Robert Price, Table of n, a(n) for n = 1..321, first 75 regions.
Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of initial terms, regions = 1..77 (2D view)
Omar E. Pol, Illustration of initial terms, regions = 1..30 (3D view)
Omar E. Pol, Visualization of regions in a diagram for A006128
Robert Price, Mathematica program to draw diagram up to n=28

Positive integers in A193870. Column 1 is A141285. Row j has length A194446(j). Row sums give A186412. Records are A000027.

Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A194436-A194439, A194447, A194448, A196025, A198381.

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 *)

Further edited by Omar E. Pol, Mar 31 2012, Jan 27 2013
Minor edits by Omar E. Pol, Apr 23 2020
Comments corrected (following a suggestion from Peter Munn) by Omar E. Pol, Jul 20 2025

A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 2, 2, 4, 2, 3, 6, 3, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 10, 5

Offset: 1

Omar E. Pol, Apr 18 2012

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

			Written as a triangle begins:
1;
2;
3;
2;
4,2;
3;
5,2,
2;
4,2;
3;
6,3,2,2;
3;
5,2;
4;
7,3,2,2;

Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285. Records give A000027. The n-th record is T(A000041(n),1).

Cf. A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A198381, A206437, A210941.

A182377 Total sum of positive ranks of all regions in the last shell of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 67, 91

Offset: 1

Omar E. Pol, Apr 29 2012

The rank of a region of n is the largest part minus the number of parts. For the definition of "region of n" see A206437. For the definition of "last shell of n" see A135010.

a(n) is also the sum of positive integers in row n of triangle A194447. First differs from A000094 at a(12).

			For n = 7 the last shell of 7 contains four regions: [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1] so we have:
----------------------------------------------------------
.        Largest    Number
Region     part    of parts    Rank
----------------------------------------------------------
.  1        3         1          2
.  2        5         2          3
.  3        4         1          3
.  4        7        15         -8
.
The sum of positive ranks is a(7) = 2 + 3 + 3 = 8.

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

Cf. A000094, A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A206437, A209616.

A196039 Total sum of the smallest part of every partition of every shell of n.

Original entry on oeis.org

0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Partial sums of A046746.

Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012

			For n = 5 the seven partitions of 5 are:
5
3         + 2
4             + 1
2     + 2     + 1
3         + 1 + 1
2     + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.

b:= proc(n, i) option remember;
     `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(nAlois P. Heinz, Apr 03 2012

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

a(n) = A066186(n) - A196025(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - Vaclav Kotesovec, Jul 06 2019

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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).

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A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

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A182377 Total sum of positive ranks of all regions in the last shell of n.

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A196039 Total sum of the smallest part of every partition of every shell of n.

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