A138137 -id:A138137 - cp's OEIS frontend

A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

0, 1, 1, 3, 2, 7, 5, 12, 13, 22, 22, 43, 43, 67, 81, 117, 133, 195, 223, 312, 373, 492, 584, 782, 925, 1190, 1433, 1820, 2170, 2748, 3268, 4075, 4872, 5997, 7150, 8781, 10420, 12669, 15055, 18198, 21535, 25925, 30602, 36624, 43201, 51428, 60478, 71802, 84215

Offset: 1

Omar E. Pol, Oct 19 2011

nonn

Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.  (8)
.  (4) + (4)
.   5  + (3)
.   6  + (2)
.   3  +  3  + (2)
.   4  + (2) + (2)
.  (2) + (2) + (2) + (2)
Note that in every partition the smallest parts are shown between parentheses. The total number of smallest parts is 1+2+1+1+1+2+4 = 12, so a(8) = 12.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n
A. Folsom and K. Ono, The spt-function of Andrews
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2020.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function
Wikipedia, Spt function

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.

b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 0, b(n, i-1)+
       add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Apr 09 2012

Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* T. D. Noe, Oct 19 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 0, b[n, i-1] + Sum[If[n== i*j, j, b[n-i*j, i-1]], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

def A195820(n):
    return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))
# D. S. McNeil, Oct 19 2011

a(n) = A092269(n) - A000070(n-1).
G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 - (72 + 5*Pi^2)*sqrt(6) / (144*Pi*sqrt(n))). - Vaclav Kotesovec, Jul 31 2017

More terms from D. S. McNeil, Oct 19 2011

A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 0, 1, 7, 1, 2, 0, 1, 9, 6, 2, 2, 0, 1, 15, 4, 4, 1, 2, 0, 1, 19, 13, 4, 5, 1, 2, 0, 1, 32, 10, 10, 3, 4, 1, 2, 0, 1, 40, 24, 10, 9, 4, 4, 1, 2, 0, 1, 60, 23, 18, 8, 8, 3, 4, 1, 2, 0, 1, 78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1

Offset: 1

Omar E. Pol, Feb 17 2012

For the calculation of row n, the number of odd/even parts, etc, take the row n from the triangle A207031 and then follow the same rules of A206563.

			Triangle begins:
  1;
  1,   1;
  3,   0,  1;
  3,   3,  0,  1;
  7,   1,  2,  0, 1;
  9,   6,  2,  2, 0, 1;
  15,  4,  4,  1, 2, 0, 1;
  19, 13,  4,  5, 1, 2, 0, 1;
  32, 10, 10,  3, 4, 1, 2, 0, 1;
  40, 24, 10,  9, 4, 4, 1, 2, 0, 1;
  60, 23, 18,  8, 8, 3, 4, 1, 2, 0, 1;
  78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1;

Cf. A006128, A066897, A066898, A135010, A138121, A138135, A138137, A141285, A181187, A182703, A206563, A207031.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A207031(n,j)).

It appears that A182703(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 1, 4, 3, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 4, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2020

T(n,k) is also the number of divisors of A336811(n,k).

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
  1;
  2;
  2, 1;
  3, 2, 1;
  2, 2, 2, 1, 1;
  4, 3, 2, 2, 2, 1, 1;
  2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1;
  4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1;
  3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of the triangle, flattened)

Number of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000005.
Row sums give A138137 (conjectured).
Cf. A135010, A138121, A138879, A187219.

A339304row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]];Array[A339304row,10] (* Paolo Xausa, Sep 01 2023 *)

a(m) = A000005(A336811(m)).

T(n,k) = A000005(A336811(n,k)).

A138136 Triangle read by rows: row n lists the parts > 1 of the last section of the set of partitions of n.

Original entry on oeis.org

0, 2, 3, 4, 2, 2, 5, 3, 2, 6, 4, 2, 3, 3, 2, 2, 2, 7, 5, 2, 4, 3, 3, 2, 2, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2, 2, 10, 8, 2, 7, 3, 6, 4, 6, 2, 2, 5, 5, 5, 3, 2, 4, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Mar 23 2008

See A138138 and A138151 for more information.

			Triangle begins:
0;
2;
3;
4,2,2;
5,3,2;
6,4,2,3,3,2,2,2;
7,5,2,4,3,3,2,2;
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2;
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2;
10,8,2,7,3,6,4,6,2,2,5,5,5,3,2,4,4,2,4,3,3,4,2,2,2,3,3,2,2,2,2,2,2,2;
...

Robert Price, Table of n, a(n) for n = 1..9514, 25 rows.

Cf. A138137, A138138, A138151.

Join[{0}, Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 10}]] // Flatten (* Robert Price, May 22 2020 *)

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 14 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First we copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.

			A table of partitions.
---------------------------------------------------------
.              Expanded       Geometric  Side view of the
Partitions     version        model      folded table
---------------------------------------------------------
1;             1;             |*|                /
---------------------------------------------------------
1,1;           1,1;           |o|*|              \
2;             . 2;           |* *|               \
---------------------------------------------------------
2,1;           . 2,1;         |o o|*|             /
1,1,1;         1,1,1;         |o|o|*|            /
3;             . . 3;         |* * *|           /
---------------------------------------------------------
3,1;           . . 3,1;       |o o o|*|         \
1,1,1,1;       1,1,1,1;       |o|o|o|*|          \
2,1,1;         . 2,1,1;       |o o|o|*|           \
2,2;           . 2,. 2;       |* *|* *|            \
4;             . . . 4;       |* * * *|             \
---------------------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|           /
2,2,1;         . 2,. 2,1;     |o o|o o|*|          /
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|         /
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|        /
3,1,1;         . . 3,1,1;     |o o o|o|*|       /
3,2;           . . 3,. 2;     |* * *|* *|      /
5;             . . . . 5;     |* * * * *|     /
---------------------------------------------------------
5,1;           . . . . 5,1;   |o o o o o|*|   \
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|    \
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|     \
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|      \
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|       \
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|        \
4,1,1;         . . . 4,1,1;   |o o o o|o|*|         \
2,2,2;         . 2, .2,. 2;   |* *|* *|* *|          \
4,2;           . . . 4,. 2;   |* * * *|* *|           \
3,3;           . . 3,. . 3;   |* * *|* * *|            \
6;             . . . . . 6;   |* * * * * *|             \
---------------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Omar E. Pol, Illustration of initial terms

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 16 2008, Mar 25 2008

The Integrated Diagram of Partitions is a shell model of partitions of a number. Partitions of n contains all partitions of the previous numbers. The number of shells of the partitions of n is equal to n. The number of parts of the last section of the set of partitions of n is A138137(n)=A006128(n)-A006128(n-1) and equal to the number of terms of row n. The number of terms of row n that are equal to 1 is A000041(n-1). The last term of row n is n. The shell model of partitions has several 2D and 3D versions.

			........................................
.. Integrated Diagram of Partitions ...
........... for n = 1 to 9 ............
.......................................
Partition number \ n = 1 2 3 4 5 6 7 8 9
........................................
.1) A000041(1)= 1 .... 1 1 1 1 1 1 1 1 1
.2) A000041(2)= 2 .... . 2 1 1 1 1 1 1 1
.3) A000041(3)= 3 .... . . 3 1 1 1 1 1 1
.4) .................. . 2 . 2 1 1 1 1 1
.5) A000041(4)= 5 .... . . . 4 1 1 1 1 1
.6) .................. . . 3 . 2 1 1 1 1
.7) A000041(5)= 7 .... . . . . 5 1 1 1 1
.8) .................. . 2 . 2 . 2 1 1 1
.9) .................. . . 3 . . 3 1 1 1
10) .................. . . . 4 . 2 1 1 1
11) A000041(6)=11 .... . . . . . 6 1 1 1
12) .................. . . 3 . 2 . 2 1 1
13) .................. . . . 4 . . 3 1 1
14) .................. . . . . 5 . 2 1 1
15) A000041(7)=15 .... . . . . . . 7 1 1
16) .................. . 2 . 2 . 2 . 2 1
17) .................. . . 3 . . 3 . 2 1
18) .................. . . . 4 . 2 . 2 1
19) .................. . . . 4 . . . 4 1
20) .................. . . . . 5 . . 3 1
21) .................. . . . . . 6 . 2 1
22) A000041(8)=22 .... . . . . . . . 8 1
23) .................. . . 3 . 2 . 2 . 2
24) .................. . . 3 . . 3 . . 3
25) .................. . . . 4 . . 3 . 2
26) .................. . . . . 5 . 2 . 2
27) .................. . . . . 5 . . . 4
28) .................. . . . . . 6 . . 3
29) .................. . . . . . . 7 . 2
30) A000041(9)=30 .... . . . . . . . . 9
.......................................
Triangle begins:
1
1,2
1,1,3,
1,1,1,2,2,4
1,1,1,1,1,2,3,5
1,1,1,1,1,1,1,2,2,2,3,3,2,4,6
1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,2,3,4,2,2,5,4,5,3,6,2,7,9

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.

Cf. A000041, A006128, A138137. See A135010 for another version.

Table[ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ Reverse@Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 5, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 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 23 2012

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

			Written as a triangle:
1;
2,1;
3,1,1;
4,2,2,1,1,1;
5,3,2,1,1,1,1,1;
6,4,3,3,2,2,2,2,1,1,1,1,1,1,1;
7,5,4,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
8,6,5,4,4,4,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

Alois P. Heinz, Rows n = 1..23, flattened

Triangle similar to A138121. Mirror of A207377. Row n has length A138137(n). Row sums give A138879. Column 1 is A000027.

Cf. A006128, A066186, A135010, A182703, A182712-A182714, A194812.

A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211999. The order of the partitions of the even integers is the same as A211989.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
2,1;           . 2,1;         |o o|*|
1,1,1;         1,1,1;         |o|o|*|
3;             . . 3;         |* * *|
--------------------------------------------
4;             . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
2,1,1;         . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
3,1;           . . 3,1;       |o o o|*|
--------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
3,2;           . . 3,. 2;     |* * *|* *|
5;             . . . . 5;     |* * * * *|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
5,1;           . . . . 5,1;   |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Other versions are A211984, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 4, 5, 3, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 5, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 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, 3, 2, 1, 1, 5, 1, 1, 4, 2, 1, 6, 1

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211989. The order of the partitions of the even integers is the same as A211999.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
1,1;           1,1;           |o|*|
2;             . 2;           |* *|
--------------------------------------------
3;             . . 3;         |* * *|
1,1,1;         1,1,1;         |o|o|*|
2,1;           . 2,1;         |o o|*|
--------------------------------------------
3,1;           . . 3,1;       |o o o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
2,1,1;         . 2,1,1;       |o o|o|*|
2,2;           . 2,. 2;       |* *|* *|
4;             . . . 4;       |* * * *|
--------------------------------------------
5;             . . . . 5;     |* * * * *|
3,2;           . . 3,. 2;     |* * *|* *|
3,1,1;         . . 3,1,1;     |o o o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
4,1;           . . . 4,1;     |o o o o|*|
--------------------------------------------
5,1;           . . . . 5,1;   |o o o o o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
1,1,1,1,1;     1,1,1,1,1,1;   |o|o|o|o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,2;         . 2,. 2,1,1;   |* *|* *|* *|
4,2;           . . . 4,1,1;   |* * * *|* *|
3,3;           . . 3,. . 3;   |* * *|* * *|
6;             . . . . . 6;   |* * * * * *|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Other versions are A211983, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211985 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged in a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then we use the same spiral of A211995.
- If the integer j is even then the first composition listed of each spiral is j.

			--------------------------------------------
.               Expanded        Geometric
Compositions   arrangement        model
--------------------------------------------
1;                 1;             |*|
--------------------------------------------
2;                 . 2;           |* *|
1,1;               1,1;           |o|*|
--------------------------------------------
3;               3 . .;         |* * *|
1,1,1;           1,1,1;         |*|o|o|
1,2;             1,. 2;         |*|o o|
--------------------------------------------
4,;              . . . 4;       |* * * *|
2,2;             . 2,. 2;       |* *|* *|
1,2,1;           1,. 2,1;       |o|o o|*|
1,1,1,1,;        1,1,1,1;       |o|o|o|*|
3,1;             3 . .,1;       |o o o|*|
--------------------------------------------
5;             5 . . . .;     |* * * * *|
2,3;           2 .,3 . .;     |* *|* * *|
1,3,1;         1,3 . .,1;     |*|o o o|o|
1,1,1,1,1;     1,1,1,1,1;     |*|o|o|o|o|
1,1,2,1;       1,1,. 2,1;     |*|o|o o|o|
1,2,2;         1,. 2,. 2;     |*|o o|o o|
1,4;           1,. . . 4;     |*|o o o o|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
1,4,1;         1,. . . 4,1;   |o|o o o o|*|
1,2,2,1;       1,. 2,. 2,1;   |o|o o|o o|*|
1,1,2,1,1;     1,1,. 2,1,1;   |o|o|o o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
1,3,1,1;       1,3 . .,1,1;   |o|o o o|o|*|
2,3,1;         2 .,3 . .,1;   |o o|o o o|*|
5,1;           5 . . . .,1;   |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Mirror of A211986. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

cp's OEIS Frontend

A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

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A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

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A138136 Triangle read by rows: row n lists the parts > 1 of the last section of the set of partitions of n.

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A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

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A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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