A238628 -id:A238628 - cp's OEIS frontend

A008967 Coefficients of Gaussian polynomials q_binomial(n-2, 2). Also triangle of distribution of rank sums: Wilcoxon's statistic. Irregular triangle read by rows.

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

Offset: 4

N. J. A. Sloane

Rows are numbers of dominoes with k spots where each half-domino has zero to n spots (in standard domino set: n=6, there are 28 dominoes and row is 1,1,2,2,3,3,4,3,3,2,2,1,1). - Henry Bottomley, Aug 23 2000
The Gaussian polynomial (or Gaussian binomial) [n,2]q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 2 by [n,2]_q = ([n]_q * [n-1]_q)/([1]_q * [2]_q), where [n]_q := q^n - 1. The first few values are: [2,2]_q = 1; [3,2]_q = 1 + q + q^2; [4,2]_q = 1 + q + 2q^2 + q^3 + q^4. - _Peter Bala, Sep 23 2007
These numbers appear in the solution of Cayley's counting problem on covariants as N(p,2,w) = [x^p,q^w] Phi(q,x) with the o.g.f. Phi(q,x) = 1/((1-x)(1-qx)(1-q^2x)) given by Peter Bala in the formula section. See the Hawkins reference, p. 264, were also references are given. - Wolfdieter Lang, Nov 30 2012
The entry a(p,w), p >= 0, w = 0,1,...,2*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 = p and 1*m_1 + 2*m_2 = w. See the Hawkins reference p. 264, (4.8). N(p,2,w) there is a(p,w). See also the Cayley reference p. 110, 35. with m = 2, Theta = p and q = w. - Wolfdieter Lang, Dec 01 2012
From Gus Wiseman, Sep 20 2023: (Start)
Also the number of unordered pairs of distinct positive integers up to n with sum k. For example, row n = 9 counts the following pairs:
  12  13  14  15  16  17  18  19  29  39  49  59  69  79  89
          23  24  25  26  27  28  38  48  58  68  78
                  34  35  36  37  47  57  67
                          45  46  56
Allowing repeated parts (x,x) gives A004737.
For strict partitions instead of just pairs we have A053632.
(End)

			1;
1,1,1;
1,1,2,1,1;
1,1,2,2,2,1,1;
1,1,2,2,3,2,2,1,1;
1,1,2,2,3,3,3,2,2,1,1;
...
Partitions: row p=2 and column w=2 has entry 2 because the 2 solutions of the two equations mentioned in a comment above are: m_0 = 0, m_1 = 2, m_2 = 0 and m_0 = 1, m_1 = 0, m_2 = 1. - _Wolfdieter Lang_, Dec 01 2012

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 242.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 236.
T. Hawkins, Emergence of the Theory of Lie Groups, Springer 2000, ch. 7.4, p. 260-5.

John Tyler Rascoe, Rows n = 4..103, flattened
A. Cayley, A Second Memoir Upon Quantics, Phil. Trans. R. Soc. London, 146 (1856) 101-126.
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.
Index entries for sequences related to dominoes

Cf. A008968, A047971, A106822.
A version with zeros is A219238.
This is the case of A365541 counting only length-2 subsets.
Cf. A000009, A004526, A004737, A053632, A068911, A167762, A238628, A365544.

qBinom := proc(n,m,q)
        mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ;
        factor(%) ;
        expand(%) ;
end proc:
A008967 := proc(n,k)
        coeftayl( qBinom(n,2,q),q=0,k ) ;
end proc:
seq(seq( A008967(n,k),k=0..2*n-4),n=2..10) ; # assumes offset 2. R. J. Mathar, Oct 13 2011

rmax = 11; f[r_] := Product[(x^i - x^(r+1))/(1-x^i), {i, 1, r-2}]/  x^((r-1)*(r-2)/2); row[r_] := CoefficientList[ Series[ f[r], {x, 0, 2rmax}], x]; Flatten[ Table[ row[r], {r, 2, rmax}]] (* Jean-François Alcover, Oct 13 2011, after given formula *)
T[n_, k_] := SeriesCoefficient[QBinomial[n - 2, 2, q], {q, 0, k}];
Table[T[n, k], {n, 4, 13}, {k, 0, 2 n - 8}] // Flatten (* Jean-François Alcover, Aug 20 2019 *)
Table[Length[Select[Subsets[Range[n],{2}],Total[#]==k&]],{n,2,15},{k,3,2n-1}] (* Gus Wiseman, Sep 20 2023 *)

print(flatten([q_binomial(n-2, 2).list() for n in (4..13)])) # Peter Luschny, Oct 23 2019

Let f(r) = Product( (x^i-x^(r+1))/(1-x^i), i = 1..r-2) / x^((r-1)*(r-2)/2); then expanding f(r) in powers of x and taking coefficients gives the successive rows of this triangle (with a different offset).
Expanding (q^n - 1)(q^(n+1) - 1)/((q - 1)(q^2 - 1)) in powers of q and taking coefficients gives the n-th row of the triangle. Ordinary generating function: 1/((1-x)(1-qx)(1-q^2x)) = 1 + x(1 + q + q^2) + x^2(1 + q + 2q^2 + q^3 + q^4) + .... - Peter Bala, Sep 23 2007
For n >= 2, let a(n,i) denote the i-th entry of the (n-1)-st row of this triangle; for every 0 <= i <= n-2, a(n,i) = a(n,2(n-2)-i) = ceiling((i+1)/2). - Christian Barrientos, Aug 08 2019

More terms from Christian Barrientos, Aug 08 2019

A086543 Number of partitions of n with at least one odd part.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 15, 17, 30, 35, 56, 66, 101, 120, 176, 209, 297, 355, 490, 585, 792, 946, 1255, 1498, 1958, 2335, 3010, 3583, 4565, 5428, 6842, 8118, 10143, 12013, 14883, 17592, 21637, 25525, 31185, 36711, 44583, 52382, 63261, 74173, 89134, 104303, 124754, 145698, 173525, 202268

Offset: 0

Vladeta Jovovic, Sep 10 2003

nonn

From Gus Wiseman, Oct 12 2023: (Start)
Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)
            (21)   (31)    (32)     (42)      (43)
            (111)  (1111)  (41)     (51)      (52)
                           (221)    (222)     (61)
                           (311)    (411)     (322)
                           (2111)   (2211)    (331)
                           (11111)  (21111)   (421)
                                    (111111)  (511)
                                              (2221)
                                              (3211)
                                              (4111)
                                              (22111)
                                              (31111)
                                              (211111)
                                              (1111111)
Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:
  (1)  (11)  (3)    (31)    (5)      (33)      (7)
             (21)   (211)   (32)     (51)      (43)
             (111)  (1111)  (41)     (321)     (52)
                            (221)    (411)     (61)
                            (311)    (2211)    (322)
                            (2111)   (3111)    (331)
                            (11111)  (21111)   (421)
                                     (111111)  (511)
                                               (2221)
                                               (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

			a(4)=3 because we have [3,1],[2,1,1] and [1,1,1] ([4] and [2,2] do not qualify).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A038348, A047967.
The complement is counted by A035363, ranks A344415.
These partitions have ranks A366322.
A025065 counts partitions with sum <= twice length, ranks A344296.
A110618 counts partitions with sum >= twice maximum, ranks A344291.
Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.

g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # Emeric Deutsch, Mar 30 2006

nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)
Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* Gus Wiseman, Oct 12 2023 *)

x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ Joerg Arndt, May 04 2013

A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).
G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - Joerg Arndt, May 04 2013

A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 7, 4, 4, 8, 8, 14, 14, 14, 8, 8, 16, 16, 28, 28, 37, 28, 28, 16, 16, 32, 32, 56, 56, 74, 74, 74, 56, 56, 32, 32, 64, 64, 112, 112, 148, 148, 175, 148, 148, 112, 112, 64, 64, 128, 128, 224, 224, 296, 296, 350, 350, 350, 296, 296, 224, 224, 128, 128

Offset: 2

Gus Wiseman, Sep 15 2023

Rows are palindromic.

			Triangle begins:
    1
    2    2    2
    4    4    7    4    4
    8    8   14   14   14    8    8
   16   16   28   28   37   28   28   16   16
   32   32   56   56   74   74   74   56   56   32   32
Row n = 4 counts the following subsets:
  {1,2}      {1,3}      {1,4}      {2,4}      {3,4}
  {1,2,3}    {1,2,3}    {2,3}      {1,2,4}    {1,3,4}
  {1,2,4}    {1,3,4}    {1,2,3}    {2,3,4}    {2,3,4}
  {1,2,3,4}  {1,2,3,4}  {1,2,4}    {1,2,3,4}  {1,2,3,4}
                        {1,3,4}
                        {2,3,4}
                        {1,2,3,4}

Row lengths are A005408.
The case counting only length-2 subsets is A008967.
Column k = n + 1 appears to be A167762.
The version for all subsets (instead of just pairs) is A365381.
Column k = n is A365544.
A000009 counts subsets summing to n.
A007865/A085489/A151897 count certain types of sum-free subsets.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A093971/A088809/A364534 count certain types of sum-full subsets.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A068911, A095944, A238628, A288728, A326083, A364272, A365376, A365377.

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#,{2}],k]&]], {n,2,11}, {k,3,2n-1}]

A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 4, 7, 8, 12, 13, 19, 21, 29, 33, 45, 49, 67, 73, 97, 108, 139, 152, 196, 217, 274, 303, 379, 420, 523, 579, 709, 786, 960, 1061, 1285, 1423, 1714, 1885, 2265, 2498, 2966, 3280, 3881, 4268, 5049, 5548, 6507, 7170, 8391, 9194, 10744, 11778, 13677

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions are necessarily incomplete (A365924).

Are there any decreases after the initial terms?

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

Chai Wah Wu, Table of n, a(n) for n = 0..65

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 partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237667 counts sum-free partitions, sum-full A237668.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000700, A124506, A238628, A240861, A364349, A364531, A365045, A365381, A365918, A366320.

Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

a(41)-a(54) from Chai Wah Wu, Nov 13 2023

A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 11, 15, 22, 30, 43, 58, 80, 106, 143, 186, 248, 318, 417, 530, 684, 863, 1103, 1379, 1741, 2162, 2707, 3339, 4145, 5081, 6263, 7640, 9357, 11350, 13822, 16692, 20214, 24301, 29300, 35073, 42085, 50208, 59981, 71294, 84866, 100509, 119206

Offset: 0

Gus Wiseman, Nov 11 2023

nonn

Or, partitions whose length is a subset-sum of the parts.

			The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)       (62)
             (111)  (211)   (221)    (321)     (322)      (332)
                    (1111)  (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

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 partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971/A364534 count certain types of sum-full subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237668 counts sum-full partitions, sum-free A237667.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, complement A366320.
A365543 counts partitions of n with a subset-sum k, strict A365661.
Cf. A000700, A238628, A363225, A364272, A365658, A365918.

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

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}]

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are the Heinz numbers of the partitions counted by A367218.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

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
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict partitions, counted by A000009.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A066208 ranks partitions into odd parts, counted by A000009.
A088809/A093971/A364534 count certain types of sum-full subsets.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A237668 counts sum-full partitions, ranks A364532.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]

A366738 Number of semi-sums of integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 17, 28, 46, 72, 111, 166, 243, 352, 500, 704, 973, 1341, 1819, 2459, 3277, 4363, 5735, 7529, 9779, 12685, 16301, 20929, 26638, 33878, 42778, 53942, 67583, 84600, 105270, 130853, 161835, 199896, 245788, 301890, 369208, 451046, 549002, 667370

Offset: 0

Gus Wiseman, Nov 06 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partitions of 6 and their a(6) = 17 semi-sums:
       (6) ->
      (51) -> 6
      (42) -> 6
     (411) -> 2,5
      (33) -> 6
     (321) -> 3,4,5
    (3111) -> 2,4
     (222) -> 4
    (2211) -> 2,3,4
   (21111) -> 2,3
  (111111) -> 2

The non-binary version is A304792.
The strict non-binary version is A365925.
For prime indices instead of partitions we have A366739.
The strict case is A366741.
A000041 counts integer partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A008967, A046663, A117855, A122768, A238628, A299701, A365543, A365544, A366753, A367095.

Table[Total[Length[Union[Total/@Subsets[#,{2}]]]&/@IntegerPartitions[n]],{n,0,15}]

More terms from Alois P. Heinz, Nov 06 2023

A366754 Number of non-knapsack integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 52, 60, 94, 118, 175, 213, 310, 373, 528, 643, 862, 1044, 1403, 1699, 2199, 2676, 3426, 4131, 5256, 6295, 7884, 9479, 11722, 14047, 17296, 20623, 25142, 29942, 36299, 43081, 51950, 61439, 73668, 87040, 103748, 122149, 145155, 170487

Offset: 0

Gus Wiseman, Nov 08 2023

nonn

A multiset is non-knapsack if there exist two different submultisets with the same sum.

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (2111)  (321)    (3211)    (422)      (3321)
                 (2211)   (22111)   (431)      (4221)
                 (3111)   (31111)   (3221)     (4311)
                 (21111)  (211111)  (4211)     (5211)
                                    (22211)    (32211)
                                    (32111)    (33111)
                                    (41111)    (42111)
                                    (221111)   (222111)
                                    (311111)   (321111)
                                    (2111111)  (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

The complement is counted by A108917, strict A275972, ranks A299702.
These partitions have ranks A299729.
The strict case is A316402.
The binary version is A366753, ranks A366740.
A000041 counts integer partitions, strict A000009.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with subset-sum k, complement A046663.
A365661 counts strict partitions with subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
Cf. A002033, A006827, A122768, A126796, A238628, A365923, A365924, A367095.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#]]&]], {n,0,15}]

a(n) = A000041(n) - A108917(n).

cp's OEIS Frontend

A008967 Coefficients of Gaussian polynomials q_binomial(n-2, 2). Also triangle of distribution of rank sums: Wilcoxon's statistic. Irregular triangle read by rows.

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A086543 Number of partitions of n with at least one odd part.

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A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

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A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

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A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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

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A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

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