A002865 -id:A002865 - cp's OEIS frontend

A138137 First differences of A006128.

1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133

Offset: 1

Omar E. Pol, Mar 18 2008

Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
From Omar E. Pol, Feb 16 2021: (Start)
Convolution of A341062 and A000041.
Convolution of A000005 and A002865.
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)

			From _Omar E. Pol_, Feb 19 2012: (Start)
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
.                                           6
.                                           3+3
.                                           4+2
.                                           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
. --- ----- ------- --------- ----------- --------------
.  1,  2,    3,      6,        8,          15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
From _Omar E. Pol_, Aug 16 2013: (Start)
The geometric model looks like this:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    2      3        6          8           15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 1 of A207031.

Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)

a(n) = A006128(n) - A006128(n-1).
a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

A008483 Number of partitions of n into parts >= 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

a(0) = 1 because the empty partition vacuously has each part >= 3. - Jason Kimberley, Jan 11 2011
Number of partitions where the largest part occurs at least three times. - Joerg Arndt, Apr 17 2011
By removing a single part of size 3, an A026796 partition of n becomes an A008483 partition of n - 3.
For n >= 3 the sequence counts the isomorphism classes of authentication codes AC(2,n,n) with perfect secrecy and with largest probability 0.5 that an interceptor could deceive with a substituted message. - E. Keith Lloyd (ekl(AT)soton.ac.uk).
For n >= 1, also the number of regular graphs of degree 2. - Mitch Harris, Jun 22 2005
(1 + 0*x + 0*x^2 + x^3 + x^4 + x^5 + 2*x^6 + ...) = (1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ...) * 1 / (1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + ...). - Gary W. Adamson, Jun 30 2009
Because the triangle A051031 is symmetric, a(n) is also the number of (n-3)-regular graphs on n vertices. Since the disconnected (n-3)-regular graph with minimum order is 2K_{n-2}, then for n > 4 there are no disconnected (n-3)-regular graphs on n vertices. Therefore for n > 4, a(n) is also the number of connected (n-3)-regular graphs on n vertices. - Jason Kimberley, Oct 05 2009
Number of partitions of n+2 such that 2*(number of parts) is a part. - Clark Kimberling, Feb 27 2014
For n >= 1, a(n) is the number of (1,1)-separable partitions of n, as defined at A239482.  For example, the (1,1)-separable partitions of 11 are [10,1], [7,1,2,1], [6,1,3,1], [5,1,4,1], [4,1,2,1,2,1], [3,1,3,1,2,1], so that a(11) = 6. - Clark Kimberling, Mar 21 2014
From Peter Bala, Dec 01 2024: (Start)
Let P(3, n) denote the set of partitions of n into parts k >= 3. Then A000041(n) = (1/2) * Sum_{parts k in all partitions in P(3, n+3)} phi(k), where phi(k) is the Euler totient function (see A000010). For example, with n = 5, there are 3 partitions of n + 3 = 8 into parts greater then 3, namely, 8, 5 + 3 and 4 + 4, and (1/2)*(phi(8) + phi(5) + phi(3) + 2*phi(4)) = 7 = A000041(5). (End)

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 301 terms from Vincenzo Librandi)
Peter Adams, Saad I. El-Zanati, Peter Florido, and William Turner, On 2-Factorizations of the Complete 3-Uniform Hypergraph of Order 12 Minus a 1-Factor, Combinatorics, Graph Theory and Computing (SEICCGTC 2021) Springer Proc. Math. Stat., Vol 448, pp. 383-392. See p. 326.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
R.-Q. Feng, J. H. Kwak and E. K. Lloyd, Isomorphism classes of authentication codes, Bull. Austral. Math. Soc. 69 (2004), no. 2, 203-215.
Elisabeth Gaar and Daniel Krenn, Metamour-regular Polyamorous Relationships and Graphs, arXiv:2005.14121 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 446
F. Jouneau-Sion and O. Torres, In Fisher's net: exact F-tests in semi-parametric models with exchangeable errors, August 2014, preprint on ResearchGate.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g.
Johan Kok, Degree affinity number of certain 2-regular graphs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
Eric Weisstein's World of Mathematics, Two-Regular Graph.

Essentially the same sequence as A026796 and A281356.
From Jason Kimberley, Nov 07 2009 and Jan 05 2011 and Feb 03 2011: (Start)
Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), this sequence (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
2-regular simple graphs: A179184 (connected), A165652 (disconnected), this sequence (not necessarily connected).
2-regular not necessarily connected graphs without multiple edges [partitions without 2 as a part]: this sequence (no loops allowed [without 1 as a part]), A027336 (loops allowed [parts may be 1]).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), this sequence (g=3), A008484 (g=4), A185325 (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10), ... (End)
Cf. A008284.

p := NumberOfPartitions; A008483 :=  func< n | n eq 0 select 1 else n le 2 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3)>; // Jason Kimberley, Jan 11 2011

series(1/product((1-x^i),i=3..50),x,51);
ZL := [ B,{B=Set(Set(Z, card>=3))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..46); # Zerinvary Lajos, Mar 13 2007
with(combstruct):ZL2:=[S,{S=Set(Cycle(Z,card>2))}, unlabeled]:seq(count(ZL2,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
with(combstruct):a:=proc(m) [A,{A=Set(Cycle(Z,card>m))},unlabeled]; end: A008483:=a(2):seq(count(A008483,size=n),n=0..46); # Zerinvary Lajos, Oct 02 2007

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 3], {n, 49}] (* Robert G. Wilson v, Jan 31 2011 *)
Rest[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Length[p]]], {n, 50}]]  (* Clark Kimberling, Feb 27 2014 *)

a(n) = numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import partition
def A008483(n): return partition(n)-partition(n-1)-partition(n-2)+partition(n-3) # Chai Wah Wu, Jun 10 2025

a(n) = p(n) - p(n - 1) - p(n - 2) + p(n - 3) where p(n) is the number of unrestricted partitions of n into positive parts (A000041).
G.f.: Product_{m>=3} 1/(1-x^m).
G.f.: (Sum_{n>=0} x^(3*n)) / (Product_{k=1..n} (1 - x^k)). - Joerg Arndt, Apr 17 2011
a(n) = A121081(n+3) - A121659(n+3). - Reinhard Zumkeller, Aug 14 2006
Euler transformation of A179184. a(n) = A179184(n) + A165652(n). - Jason Kimberley, Jan 05 2011
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Feb 26 2015
G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
a(n) = Sum_{j=0..floor(n/2)} A008284(n-2*j,j). - Gregory L. Simay, Apr 27 2023
G.f.: 1 + Sum_{n >= 1} x^(n+2)/Product_{k = 0..n-1} (1 - x^(k+3)). - Peter Bala, Dec 01 2024

A237113 Number of partitions of n such that some part is a sum of two other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326

Offset: 0

Clark Kimberling, Feb 04 2014

nonn

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 3.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(0) = 0 through a(9) = 10 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (21111)  (211111)  (3221)     (4311)
                                                (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (42111)
                                                (221111)   (222111)
                                                (2111111)  (321111)
                                                           (2211111)
                                                           (21111111)
(End)

The complement for subsets is A085489, with re-usable parts A007865.
For subsets of {1..n} we have A088809, with re-usable parts A093971.
The complement is counted by A236912, ranks A364461.
The non-binary complement is A237667, ranks A364531.
The non-binary version is A237668, ranks A364532.
With re-usable parts we have A363225, ranks A364348.
The complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364462.
The strict case is A364670, with re-usable parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A151897, A237984, A325862, A326083, A363260.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,30}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A236912(n).

a(0)=0 prepended by Alois P. Heinz, Sep 17 2023

A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 1, 8, 6, 3, 2, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 1, 14, 11, 5, 4, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 21 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			Triangle begins:
  1
  0  1
  1  0  1
  1  0  0  1
  2  1  0  0  1
  2  2  0  0  0  1
  4  2  1  0  0  0  1
  4  3  2  0  0  0  0  1
  7  4  3  1  0  0  0  0  1
  8  6  3  2  0  0  0  0  0  1
 12  8  4  3  1  0  0  0  0  0  1
 14 11  5  4  2  0  0  0  0  0  0  1
 21 14  8  4  3  1  0  0  0  0  0  0  1
 24 20 10  5  4  2  0  0  0  0  0  0  0  1
 34 25 15  6  5  3  1  0  0  0  0  0  0  0  1
For example, row n = 9 counts the following partitions:
  (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
  (3,3,1,1,1)          (6,2,1)      (4,3,2)
  (4,2,1,1,1)          (2,2,2,2,1)  (5,3,1)
  (5,1,1,1,1)          (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)

Column k=1 is A002865(n-1).
Row sums are A027193 (odd-length ptns), strict A067659.
This is the odd-length case of A359901, with half-steps A359893.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
A325347 counts partitions w/ integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
Cf. A008289, A026424, A327472, A359889, A359895, A359906, A359907, A359910.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Median[#]==k&]],{n,15},{k,n}]

A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701, 35717

Offset: 1

Omar E. Pol, Dec 09 2011

nonn

Essentially the same as A002865, but here a(1) = 1 not 0.
Also number of regions in the last section of the set of partitions of n.
Also number of partitions of n+k that are formed by k+1 sections, k >= 0 (Cf. A194799). - Omar E. Pol, Jan 30 2012
For the definition of region see A206437. - Omar E. Pol, Aug 13 2013
Partial sums give A000041, n >= 1. - Omar E. Pol, Sep 04 2013
Also the number of partitions of n with no parts greater than the number of ones. - Spencer Miller, Jan 28 2023

			From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms as number of regions:
.                                           _ _ _ _ _ _
.                                          |_ _ _      |
.                                          |_ _ _|_    |
.                                          |_ _    |   |
.                               _ _ _ _ _  |_ _|_ _|_  |
.                              |_ _ _    |           | |
.                     _ _ _ _  |_ _ _|_  |           | |
.                    |_ _    |         | |           | |
.             _ _ _  |_ _|_  |         | |           | |
.       _ _  |_ _  |       | |         | |           | |
.   _  |_  |     | |       | |         | |           | |
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1     1       1         2           2             4
.
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A002865, A083751, A135010, A186114, A193870, A206437, A225600, A225610.

Join[{1}, Drop[CoefficientList[Series[1 / Product[(1 - x^k)^1, {k, 2, 50}], {x, 0, 50}], x], 2]] (* Vincenzo Librandi, Feb 15 2018 *)
A187219[nmax_]:=Join[{1},Differences[PartitionsP[Range[nmax]]]];
A187219[100] (* Paolo Xausa, Feb 17 2023 *)

a(n) = A083751(n) + 1. - Omar E. Pol, Mar 04 2012

a(n) = A002865(n), if n >= 2. - Omar E. Pol, Aug 13 2013

Better definition from Omar E. Pol, Sep 04 2013

A362608 Number of integer partitions of n having a unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 11, 16, 21, 29, 43, 54, 78, 102, 131, 175, 233, 295, 389, 490, 623, 794, 1009, 1255, 1579, 1967, 2443, 3016, 3737, 4569, 5627, 6861, 8371, 10171, 12350, 14901, 18025, 21682, 26068, 31225, 37415, 44617, 53258, 63313, 75235, 89173, 105645

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For parts instead of multiplicities we have A000041(n-1), ranks A102750.
For median instead of mode we have A238478, complement A238479.
These partitions have ranks A356862.
The complement is counted by A362607, ranks A362605.
For co-mode complement we have A362609, ranks A362606.
For co-mode we have A362610, ranks A359178.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A237984, A304442, A327472, A360071, A360687, A362612.

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]==1&]],{n,0,30}]

seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - Andrew Howroyd, May 04 2023

A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 1, 4, 1, 2, 1, 30, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 42, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 56, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 26 2011

For the definition of "region" of the set of partitions of j, see A206437.
a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.
Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.
a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012
From Omar E. Pol, Aug 19 2013: (Start)
In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).
a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.
(End)
The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

			Written as an irregular triangle the sequence begins:
  1;
  2;
  3;
  1, 5;
  1, 7;
  1, 2, 1, 11;
  1, 2, 1, 15;
  1, 2, 1,  4, 1, 1, 22;
  1, 2, 1,  4, 1, 2,  1, 30;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;
  1, 2, 1,  4, 1, 2,  1,  8, 1, 1, 3,  1, 1, 56;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 12, 1,  2, 1, 4, 1, 2, 1, 1, 77;
  ...
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     2       3     1         5       1           7
.
The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).
.      _ _ _ _ _
.  7   _ _ _    |
.  6   _ _ _|_  |
.  5   _ _    | |
.  4   _ _|_  | |
.  3   _ _  | | |
.  2   _  | | | |
.  1    | | | | |
.
.      1 2 3 4 5
.
Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.
.                                    /\
.                                   /  \
.                      /\          /    \
.                     /  \        /      \
.            /\      /    \    /\/        \
.       /\  /  \  /\/      \  / 1          \
.    /\/  \/    \/ 1        \/              \
.     1   2     3           5               7
.
(End)

Robert Price, Table of n, a(n) for n = 1..5603
Omar E. Pol, Illustration of the seven regions of 5

Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137.

Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A194446 = {}; l = {};
For[j = 1, j <= 30, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A194446, j - i];
  ];
A194446   (* Robert Price, Jul 25 2020 *)

a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012

A319000 Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 3, 3, 3, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 22 2018

			Triangle begins:
  1
  0 1
  0 0 1
  0 0 0 2
  0 0 0 0 1
  0 0 0 0 1 2
  0 0 0 0 0 0 1
  0 0 0 0 0 2 2 3
  0 0 0 0 0 1 1 1 2
  0 0 0 0 0 0 1 1 1 2
  0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 2 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 1 1 1 1 1 2
  0 0 0 0 0 0 0 1 1 1 1 1 1 1 2
  0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 5
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 1 2 2 3 3 3 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 2 2 2 3 3 3 3 3 3 3 3 4
Row 12 {0,0,0,0,0,0,2,3,3,3,3,4} corresponds to the partitions (C = 12):
. . . . . . (43)  (62)   (621)   (6211)   (62111)    (C)
            (322) (431)  (4311)  (43111)  (431111)   (621111)
                  (3221) (32211) (322111) (3221111)  (4311111)
                                          (32211111)

Row sums are A319916. Column sums are A319005. Last column is A001055.

Cf. A000041, A002865, A069016, A096276, A301987, A318950, A319057.

Table[Length[Select[IntegerPartitions[k],Times@@#==n&]],{n,20},{k,n}]

A236912 Number of partitions of n such that no part is a sum of two other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 12, 14, 20, 25, 34, 40, 54, 64, 85, 98, 127, 149, 189, 219, 277, 316, 395, 456, 557, 638, 778, 889, 1070, 1226, 1461, 1667, 1978, 2250, 2645, 3019, 3521, 3997, 4652, 5267, 6093, 6909, 7943, 8982, 10291, 11609, 13251, 14947, 16984, 19104

Offset: 0

Clark Kimberling, Feb 01 2014

nonn

These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - Gus Wiseman, Aug 09 2023

			Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1].  Thus, a(6) = 11 - 3 = 8.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (3111)    (421)      (521)
                                     (111111)  (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)
(End)

For subsets of {1..n} we have A085489, complement A088809.
The complement is counted by A237113, ranks A364462.
The non-binary version is A237667, ranks A364531.
The non-binary complement is A237668, ranks A364532.
The version with re-usable parts is A364345, ranks A364347.
The (strict) version for linear combinations of parts is A364350.
These partitions have ranks A364461.
The strict case is A364533, non-binary A364349.
The strict complement is A364670, with re-usable parts A363226.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A237113(n).

a(0)=1 prepended by Alois P. Heinz, Sep 17 2023

A237667 Number of partitions of n such that no part is a sum of two or more other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 61, 61, 87, 85, 120, 117, 160, 156, 224, 216, 288, 277, 380, 363, 483, 474, 622, 610, 783, 755, 994, 986, 1235, 1191, 1549, 1483, 1876, 1865, 2306, 2279, 2806, 2732, 3406, 3413, 4091, 4013, 4991, 4895, 5872

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

From Gus Wiseman, Aug 09 2023: (Start)
Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).
These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.
(End)

			For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
From _Gus Wiseman_, Aug 09 2023: (Start)
The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

For subsets of {1..n} we have A151897, binary A085489.
The binary version is A236912, ranks A364461.
The binary complement is A237113, ranks A364462.
The complement is counted by A237668, ranks A364532.
The binary version with re-usable parts is A364345, strict A364346.
The strict case is A364349, binary A364533.
These partitions have ranks A364531.
The complement for subsets is A364534, binary A088809.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A007865, A179009, A325862, A326083, A363225, A363226, A364348, A364350, A364670.

Map[Count[Map[MemberQ[#,Apply[Alternatives,Map[Apply[Plus,#]&, DeleteDuplicates[DeleteCases[Subsets[#],?(Length[#]<2&)]]]]]&, IntegerPartitions[#]],False]&,Range[20]]  (* _Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

a(21)-a(53) from Giovanni Resta, Feb 22 2014

cp's OEIS Frontend

A138137 First differences of A006128.

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A008483 Number of partitions of n into parts >= 3.

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A237113 Number of partitions of n such that some part is a sum of two other parts.

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A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

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A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

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A362608 Number of integer partitions of n having a unique mode.

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A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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