A188431 -id:A188431 - cp's OEIS frontend

A126796 Number of complete partitions of n.

1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 31, 39, 55, 71, 100, 125, 173, 218, 291, 366, 483, 600, 784, 971, 1244, 1538, 1957, 2395, 3023, 3693, 4605, 5604, 6942, 8397, 10347, 12471, 15235, 18309, 22267, 26619, 32219, 38414, 46216, 54941, 65838, 77958, 93076, 109908

Offset: 0

Brian Hopkins, Feb 20 2007

A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.
A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters, Mar 22 2007
For n > 0: a(n) = sum of n-th row in A261036 and also a(floor(n/2)) = A261036(n,floor((n+1)/2)). - Reinhard Zumkeller, Aug 08 2015
a(n+1) is the number of partitions of n such that each part is no more than 2 more than the sum of all smaller parts (generalizing Adams-Watters's criterion). Bijection: each partition counted by a(n+1) must contain a 1, removing that gives a desired partition of n. - Brian Hopkins, May 16 2017
A partition (x_1, ..., x_k) is complete if and only if 1, x_1, ..., x_k is a "regular sequence" (see A003513 for definition). As a result, the number of complete partitions with n parts is given by A003513(n+1). - Nathaniel Johnston, Jun 29 2023

			There are a(5) = 4 complete partitions of 5:
  [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [1, 1, 3].
G.f.: 1 = 1*(1-x) + 1*x*(1-x)*(1-x^2) + 1*x^2*(1-x)*(1-x^2)*(1-x^3) + 2*x^3*(1-x)*(1-x^2)*(1-x^3)*(1-x^4) + 2*x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5) + ...
From _Gus Wiseman_, Oct 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (11)  (21)   (211)   (221)    (321)     (421)      (3221)
             (111)  (1111)  (311)    (2211)    (2221)     (3311)
                            (2111)   (3111)    (3211)     (4211)
                            (11111)  (21111)   (4111)     (22211)
                                     (111111)  (22111)    (32111)
                                               (31111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 301 terms from David W. Wilson)
George E. Andrews, George Beck, and Brian Hopkins, On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions, Annals of Comb., 24(2020), pp. 217-224.
George Beck and Shane Chern, Reciprocity between partitions and compositions, arXiv:2108.04363 [math.CO], 2021.
Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.
SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Cf. A002033, A003513, A209405, A261036, A286929, A286097.
For parts instead of sums we have A000009 (sc. coverings), ranks A055932.
The strict case is A188431, complement A365831.
These partitions have ranks A325781.
First column k = 0 of A365923.
The complement is counted by A365924, ranks A365830.
Cf. A000041, A018818, A046663, A047967, A276024, A304792, A325799, A365543, A365658, A365918, A365921.

import Data.MemoCombinators (memo3, integral, Memo)
a126796 n = a126796_list !! n
a126796_list = map (pMemo 1 1) [0..] where
   pMemo = memo3 integral integral integral p
   p   0 = 1
   p s k m
     | k > min m s = 0
     | otherwise   = pMemo (s + k) k (m - k) + pMemo s (k + 1) m
-- Reinhard Zumkeller, Aug 07 2015

isCompl := proc(p,n) local m,pers,reps,f,lst,s; reps := {}; pers := combinat[permute](p); for m from 1 to nops(pers) do lst := op(m,pers); for f from 1 to nops(lst) do s := add( op(i,lst),i=1..f); reps := reps union {s}; od; od; for m from 1 to n do if not m in reps then RETURN(false); fi; od; RETURN(true); end: A126796 := proc(n) local prts, res,p; prts := combinat[partition](n); res :=0; for p from 1 to nops(prts) do if isCompl(op(p,prts),n) then res := res+1; fi; od; RETURN(res); end: for n from 1 to 40 do printf("%d %d ",n,A126796(n)); od; # R. J. Mathar, Feb 27 2007
# At the beginning of the 2nd Maple program replace the current 15 by any other positive integer n in order to obtain a(n). - Emeric Deutsch, Mar 04 2007
with(combinat): a:=proc(n) local P,b,k,p,S,j: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: end: seq(a(n),n=1..30); # Emeric Deutsch, Mar 04 2007
with(combinat): n:=15: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: b; # Emeric Deutsch, Mar 04 2007

T[n_, k_] := T[n, k] = If[k <= 1, 1, If[n < 2k-1, T[n, Floor[(n+1)/2]], T[n, k-1] + T[n-k, k]]];
a[n_] := T[n, Floor[(n+1)/2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 11 2017, after Franklin T. Adams-Watters *)
nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]]; Table[Length[Select[IntegerPartitions[n], nmz[#]=={}&]],{n,0,15}] (* Gus Wiseman, Oct 14 2023 *)

{T(n,k)=if(k<=1,1,if(n<2*k-1,T(n,floor((n+1)/2)),T(n,k-1)+T(n-k,k)))}
{a(n)=T(n,floor((n+1)/2))} /* If modified to save earlier results, this would be efficient. */ /* Franklin T. Adams-Watters, Mar 22 2007 */

/* As coefficients in g.f.: */
{a(n)=local(A=[1,1]);for(i=1,n+1,A=concat(A,0);A[#A]=polcoeff(1-sum(m=1,#A,A[m]*x^m*prod(k=1,m,1-x^k +x*O(x^#A))),#A) );A[n+1]}
for(n=0,50,print1(a(n),",")) /* Paul D. Hanna, Mar 06 2012 */

G.f.: 1 = Sum_{n>=0} a(n)*x^n*Product_{k=1..n+1} (1-x^k). - Paul D. Hanna, Mar 08 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n) + (25/16 - 1679*Pi^2/6912)/n). - Vaclav Kotesovec, May 24 2018, extended Nov 02 2019
a(n) = A000041(n) - A365924(n). - Gus Wiseman, Oct 14 2023

More terms from R. J. Mathar, Feb 27 2007
More terms from Emeric Deutsch, Mar 04 2007
Further terms from Franklin T. Adams-Watters and David W. Wilson, Mar 22 2007

A325781 Heinz numbers of complete integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 160, 162, 168, 176, 180, 192, 198, 200, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288, 294, 300

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff its divisors have sums of prime indices covering an initial interval of nonnegative integers. For example, the divisors of 60 are {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, with respective sums of prime indices {0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7}, so 60 is in the sequence.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    30: {1,2,3}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    42: {1,2,4}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    64: {1,1,1,1,1,1}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A325684, A325770, A325780, A325782.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],normQ[hwt/@Rest[Divisors[#]]]&]

A326020 Number of complete subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 15, 27, 50, 95, 185, 365, 724, 1441, 2873, 5735, 11458, 22902, 45789, 91561, 183102, 366180, 732331, 1464626, 2929209, 5858367, 11716674, 23433277, 46866473, 93732852, 187465596, 374931067, 749861989, 1499723808, 2999447418

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,3,4}
                                      {1,2,3,5}    {1,2,3,5}
                                      {1,2,4,5}    {1,2,3,6}
                                      {1,2,3,4,5}  {1,2,4,5}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,4,5,6}
                                                   {1,2,3,4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326021, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,0,10}]

a(17)-a(34) from Charlie Neder, Jun 05 2019

A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

The complement (complete partitions) is A126796.

			The a(0) = 0 through a(8) = 12 partitions:
  .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
                  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
                  (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
                                (5,1)    (6,1)    (6,2)
                                (2,2,2)  (3,2,2)  (7,1)
                                (4,1,1)  (3,3,1)  (3,3,2)
                                         (5,1,1)  (4,2,2)
                                                  (4,3,1)
                                                  (5,2,1)
                                                  (6,1,1)
                                                  (2,2,2,2)
                                                  (5,1,1,1)

Joerg Arndt, Table of n, a(n) for n = 0..10000

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
The complement is A126796, ranks A325781, strict A188431.
These partitions have ranks A365830.
The strict case is A365831.
Row sums of A365923 without the first column, strict A365545.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

a(n) = A000041(n) - A126796(n).

A325780 Heinz numbers of perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}

Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).

A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

Offset: 0

Gus Wiseman, Sep 28 2023

nonn

			The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
The a(2) = 1 through a(10) = 9 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
            (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
                   (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
                                 (6,1)  (7,1)    (7,2)    (8,2)
                                        (4,3,1)  (8,1)    (9,1)
                                        (5,2,1)  (4,3,2)  (5,3,2)
                                                 (5,3,1)  (5,4,1)
                                                 (6,2,1)  (6,3,1)
                                                          (7,2,1)

For parts instead of sums we have ranks A080259, A055932.
The strict complement is A188431, non-strict A126796 (ranks A325781).
Row sums of A365545 without the first column, non-strict A365923.
The non-strict version is A365924, ranks A365830.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[nmz[#]]>0&]],{n,0,15}]

A365925 Number of subset-sums of strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 17, 22, 29, 42, 59, 74, 102, 130, 171, 226, 281, 356, 454, 566, 699, 896, 1080, 1342, 1637, 2006, 2413, 2962, 3548, 4286, 5114, 6148, 7272, 8738, 10268, 12224, 14387, 16996, 19863, 23450, 27257, 31984, 37187, 43364, 50173, 58428, 67322

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

This is the "not necessarily positive" version, cf. A284640.

			The a(6) = 17 ways, showing each strict partition and its subset-sums:
    (6): 0,6
   (51): 0,1,5,6
   (42): 0,2,4,6
  (321): 0,1,2,3,4,5,6

The positive case is A284640.
The non-strict version is A304792, positive case A276024.
Row sums of A365661, non-strict A365543.
The complement (non-subset-sums) is A365922, non-strict A365918.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A365923 counts partitions by non-subset-sums, strict A365545.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A046663, A364272, A364350, A365658, A365663, A365921.

Table[Total[Length[Union[Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367224 /\ A005117.

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (7,2,1)
                                                               (4,3,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214*   A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A275972 counts strict knapsack partitions, non-strict A108917.
A364272 counts sum-full strict partitions, sum-free A364349.
A365925 counts subset-sums of strict partitions, non-strict A304792.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367225 /\ A005117.

			The a(2) = 1 through a(11) = 7 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (10)     (11)
            (3,1)  (4,1)  (5,1)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                                 (6,1)  (7,1)  (6,3)  (7,3)    (7,4)
                                               (8,1)  (9,1)    (8,3)
                                                      (5,4,1)  (10,1)
                                                               (5,4,2)
                                                               (6,4,1)
The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15):
  2  3  4   5   6   7   8   9   A    B    C    D    E     F
        31  41  51  43  53  54  64   65   75   76   86    87
                    61  71  63  73   74   84   85   95    96
                            81  91   83   93   94   A4    A5
                                541  A1   B1   A3   B3    B4
                                     542  642  C1   D1    C3
                                     641  651  652  752   E1
                                          741  742  761   654
                                               751  842   762
                                               841  851   852
                                                    941   861
                                                    6521  942
                                                          951
                                                          A41
                                                          7521

Chai Wah Wu, Table of n, a(n) for n = 0..118

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215*   A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A007865/A085489/A151897 count certain types of sum-free subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A237667 counts sum-free partitions, ranks A364531.
A240861 counts strict partitions with length not a part, complement A240855.
A275972 counts strict knapsack partitions, non-strict A108917.
A364349 counts sum-free strict partitions, sum-full A364272.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365663 counts strict partitions without a subset-sum k, non-strict A046663.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367218.

			The a(3) = 1 through a(10) = 7 strict partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                (4,1)  (5,1)    (6,1)    (7,1)    (8,1)    (9,1)
                       (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                         (5,2,1)  (5,3,1)  (5,4,1)
                                                  (6,2,1)  (6,3,1)
                                                           (7,2,1)
                                                           (4,3,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220*   A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
A365046 counts combination-full subsets, differences of A364914.
Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]

cp's OEIS Frontend

A126796 Number of complete partitions of n.

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A325781 Heinz numbers of complete integer partitions.

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A326020 Number of complete subsets of {1..n}.

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A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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A325780 Heinz numbers of perfect integer partitions.

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A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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A365925 Number of subset-sums of strict integer partitions of n.

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A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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