A138136 -id:A138136 - cp's OEIS frontend

A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

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, 5, 5, 6, 2, 2, 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, Nov 29 2010

In this sequence a(68)=5 but in A138136 a(68)=6. See the 8th term in row 10 of triangle.

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

Cf. A135010, A138121, A138136, A138151, A182710.

Row sums give A138880.

A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 2, 3, 8, 10, 24, 28, 56, 72, 120, 154, 252, 312, 476, 615, 880, 1122, 1584, 1995, 2740, 3465, 4620, 5819, 7680, 9575, 12428, 15498, 19824, 24563, 31170, 38378, 48224, 59202, 73678, 90055, 111384, 135420, 166364, 201630, 246120, 297045, 360822

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Sum of all parts > 1 of the last section of the set of partitions of n.
Row sums of triangle A182710. Also row sums of other similar triangles as A138136 and A182711.
Partial sums give A194552. - Omar E. Pol, Sep 23 2013

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A066186, A133041, A138135, A138136, A138137, A138138, A138151, A138879, A139100.

Table[Total[Flatten[Select[IntegerPartitions[n],FreeQ[#,1]&]]],{n,50}] (* Harvey P. Dale, May 24 2015 *)
a[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = A002865(n)*n = (A000041(n) - A000041(n-1))*n = A138879(n) - A000041(n-1).
a(n) ~ Pi^2/6*A000070(n-2). - Peter Bala, Dec 23 2013
G.f.: x*f'(x), where f(x) = Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Jul 06 2019

Better definition from Omar E. Pol, Sep 23 2013

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

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

Offset: 1

Omar E. Pol, Mar 21 2008

The remainder of row n is necessarily A000041(n-1) 1's.
Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
Row n is also the n-th row of A138138 in reverse order.

			Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.
Omar E. Pol, Illustration of the rows 1..6 of the triangle

Mirror of A138138.
Row lengths give A138137.
Row sums give A138879.
Column 1 gives A000027.
Right border gives A000012.
Another version is A138121 which is the mirror of A135010.
Cf. A000041, A080577, A138136, A139094, A139100.

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

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 15 2008

See the integrated diagram of partitions in the entry A138138.
See A138151 for more information.
First 43 members = A026792.

			Triangle begins:
{(1)}
{(2), (1, 1)}
{(3), (2, 1), (1, 1, 1)}
{(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)}
{(5), (3, 2), (4, 1), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1)}

Robert Price, Table of n, a(n) for n = 1..7043, 17 rows.

Cf. A000041, A006128, A026792, A036036, A080576, A080577, A135010, A138121, A138135, A138136, A138137, A138138, A138151.

Table[If[n == 1, ConstantArray[{1}, i - n + 1],
   Map[(Join[#, ConstantArray[{1}, i - n]]) &,
    Cases[IntegerPartitions[n], x_ /; Last[x] != 1]]], {i, 7}, {n, i, 1, -1}]  // Flatten(* Robert Price, May 28 2020 *)

A176210 Triangle read by rows in which row n (n>=3) lists those partitions of n with every part > 2.

Original entry on oeis.org

3, 4, 5, 6, 3, 3, 7, 4, 3, 8, 5, 3, 4, 4, 9, 6, 3, 5, 4, 3, 3, 3, 10, 7, 3, 6, 4, 5, 5, 4, 3, 3, 11, 8, 3, 7, 4, 6, 5, 5, 3, 3, 4, 4, 3, 12, 9, 3, 8, 4, 7, 5, 6, 6, 6, 3, 3, 5, 4, 3, 4, 4, 4, 3, 3, 3, 3, 13, 10, 3, 9, 4, 8, 5, 7, 6, 7, 3, 3, 6, 4, 3, 5, 5, 3, 5, 4, 4, 4, 3, 3, 3

Offset: 3

Vladimir Shevelev, Apr 12 2010

Each partition is listed in nonincreasing order.
The partitions in each row are listed in decreasing lexicographic order.
Also the numbers of vertices of the connected components of the 2-regular simple graphs on n vertices.

			For n in {0,1,2} there are no parts; so those rows are empty.
3 (one partition only)
4 (one partition only)
5 (one partition only)
6; 3, 3
7; 4, 3
8; 5, 3; 4, 4
9; 6, 3; 5, 4; 3, 3, 3
10; 7, 3; 6, 4; 5, 5; 4, 3, 3
11; 8, 3; 7, 4; 6, 5; 5, 3, 3; 4, 4, 3
12; 9, 3; 8, 4; 7, 5; 6, 6; 6, 3, 3; 5, 4, 3; 4, 4, 4; 3, 3, 3, 3
13; 10, 3; 9, 4; 8, 5; 7, 6; 7, 3, 3; 6, 4, 3; 5, 5, 3; 5, 4, 4; 4, 3, 3, 3

J. S. Kimberley, Rows 3..39 of A176210 triangle, flattened [b-file corrected by _N. J. A. Sloane_, Oct 05 2010]

The number of partitions in each row is A008483.
The length of each row is A177739.
The same ordering is used in A080577 and A138136 (for other orderings see A036036 and A036037).

&cat[ &cat RestrictedPartitions(n,{3..n}):n in [1..13]];

Extensively edited by Jason Kimberley, May 13 2010

A176211 Numbers of the form Product_{m_i >= 3} A000211(m_i), possibly repeated, in natural order.

Original entry on oeis.org

6, 9, 13, 20, 31, 36, 49, 54, 78, 78, 81, 117, 120, 125, 169, 180, 186, 201, 216, 260, 279, 294, 324, 324, 400, 403, 441, 468, 468, 486, 523, 620, 637, 702, 702, 720, 729, 750, 845, 961, 980, 1014, 1014, 1053, 1080, 1116, 1125, 1206, 1296, 1366, 1519, 1521, 1560, 1560, 1620, 1625, 1674, 1764, 1809, 1944, 1944, 2197, 2209

Offset: 1

Vladimir Shevelev, Apr 12 2010

nonn

Values represented by more than one set of indices are listed once per set; otherwise A176212 results.
Each term is a permanent of a quadratic symmetric (0,1)  matrix with 1's on the main diagonal and exactly three 1's in each row and column.
For fixed Sum m_i=n with m_i >= 3, Product A000211(m_i) >= 6(4/3)^(n-3) and max(Product A000211(m_i)) = 6^((n-h)/3)*floor((3/2)^h), where h is the remainder of n (mod 3).

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317.

Cf. A176210, A036036, A036037, A080577, A138136, A000211.

f(n) = fibonacci(n+1) + fibonacci(n-1) + 2; \\ A000211
lista(nn) = {my(v = vector(nn, k, f(k+2))); my(vmax = vecmax(v)); my(w =  vector(nn, k, [0, logint(vmax, v[k])])); my(list=List()); forvec(x = w, if (vecmax(x), my(y = prod(k=1, #v, v[k]^x[k])); if (y <= vmax, listput(list, y)););); Vec(vecsort(list));}
lista(14) \\ Michel Marcus, Jan 06 2021

A182710 Triangle read by rows in which row n lists the parts >= 2 of the last section of the set of partitions of n, in the order produced by the shell model of partitions of A138121, with a(1)=0.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 28 2010

These are the parts located in the "head" of the last section.

For other versions see A138136 and A182711.

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

Omar E. Pol, Illustration of the shell model of partitions (3D view)
Omar E. Pol, Illustration. How to build the last section of the set of partitions of n: copy, paste and fill.

Cf. A135010, A138121, A138136, A182711.

A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 3, 4, 6, 3, 4, 5, 7, 2, 3, 4, 4, 5, 6, 8, 3, 3, 4, 5, 5, 6, 7, 9, 2, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 11, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 12

Offset: 1

Omar E. Pol, May 26 2008

nonn

Omar E. Pol, A shell model of partitions
Omar E. Pol, Illustration of initial terms

Cf. A138135, A138136, A138137, A138138, A138151, A138879, A138880, A139100.

cp's OEIS Frontend

A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

Views

Author

Keywords

Comments

Examples

Crossrefs

A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A176210 Triangle read by rows in which row n (n>=3) lists those partitions of n with every part > 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Extensions

A176211 Numbers of the form Product_{m_i >= 3} A000211(m_i), possibly repeated, in natural order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A182710 Triangle read by rows in which row n lists the parts >= 2 of the last section of the set of partitions of n, in the order produced by the shell model of partitions of A138121, with a(1)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

Views

Author

Keywords

Links

Crossrefs