A365541 -id:A365541 - cp's OEIS frontend

A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 3, 3, 5, 7, 5, 5, 5, 5, 7, 11, 7, 8, 6, 8, 7, 11, 15, 11, 11, 11, 11, 11, 11, 15, 22, 15, 17, 15, 14, 15, 17, 15, 22, 30, 22, 23, 23, 22, 22, 23, 23, 22, 30, 42, 30, 33, 30, 33, 25, 33, 30, 33, 30, 42

Offset: 0

Gus Wiseman, Sep 16 2023

Rows are palindromic.

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   3   3   5
   7   5   5   5   5   7
  11   7   8   6   8   7  11
  15  11  11  11  11  11  11  15
  22  15  17  15  14  15  17  15  22
  30  22  23  23  22  22  23  23  22  30
  42  30  33  30  33  25  33  30  33  30  42
  56  42  45  44  44  43  43  44  44  45  42  56
  77  56  62  58  62  56  53  56  62  58  62  56  77
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (321)     (411)     (411)     (51)
  (42)      (321)     (321)     (3111)    (321)     (321)     (42)
  (411)     (3111)    (3111)    (2211)    (3111)    (3111)    (411)
  (33)      (2211)    (222)     (21111)   (222)     (2211)    (33)
  (321)     (21111)   (2211)    (111111)  (2211)    (21111)   (321)
  (3111)    (111111)  (21111)             (21111)   (111111)  (3111)
  (222)               (111111)            (111111)            (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Robert Price, Table of n, a(n) for n = 0..324

Columns k = 0 and k = n are A000041.
Central column n = 2k is A002219.
The complement is counted by A046663, strict A365663.
Row sums are A304792.
For subsets instead of partitions we have A365381.
The strict case is A365661.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A088809, A093971, A122768, A108917, A299701, A364911, A365541, A365658.

Table[Length[Select[IntegerPartitions[n],MemberQ[Total/@Subsets[#],k]&]],{n,0,15},{k,0,n}]

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset: 0

Gus Wiseman, Sep 16 2023

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?

			Triangle begins:
  1
  1  1
  1  0  1
  2  1  1  2
  2  1  0  1  2
  3  1  1  1  1  3
  4  2  2  1  2  2  4
  5  2  2  2  2  2  2  5
  6  3  2  3  1  3  2  3  6
  8  3  3  4  3  3  4  3  3  8
Row n = 6 counts the following strict partitions:
  (6)      (5,1)    (4,2)    (3,2,1)  (4,2)    (5,1)    (6)
  (5,1)    (3,2,1)  (3,2,1)           (3,2,1)  (3,2,1)  (5,1)
  (4,2)                                                 (4,2)
  (3,2,1)                                               (3,2,1)
Row n = 10 counts the following strict partitions:
  A     91    82    73    64    532   64    73    82    91    A
  64    541   532   532   541   541   541   532   532   541   64
  73    631   721   631   631   4321  631   631   721   631   73
  82    721   4321  721   4321        4321  721   4321  721   82
  91    4321        4321                    4321        4321  91
  532                                                         532
  541                                                         541
  631                                                         631
  721                                                         721
  4321                                                        4321

Robert Price, Table of n, a(n) for n = 0..1325

Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

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

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

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.

Original entry on oeis.org

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

A068911 Number of n-step walks (each step +-1 starting from 0) which are never more than 2 or less than -2.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 54, 108, 162, 324, 486, 972, 1458, 2916, 4374, 8748, 13122, 26244, 39366, 78732, 118098, 236196, 354294, 708588, 1062882, 2125764, 3188646, 6377292, 9565938, 19131876, 28697814, 57395628, 86093442, 172186884, 258280326, 516560652

Offset: 0

Henry Bottomley, Mar 06 2002

From Johannes W. Meijer, May 29 2010: (Start)
a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n >= 0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).
Counts all paths of length n, n >= 0, starting at the third node on the path graph P_5, see the Maple program. (End)
From Alec Jones, Feb 25 2016: (Start)
The a(n) are the n-th terms in a "Fibonacci snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.)
For example:
  1 ...  1      1  ...   1 4      1 4 6 ...  1 4 6       1 4 6   ...  and so on.
         1 ...  1 2      1 2 ...  1 2        1 2 12 ...  1 2 12 18 (End)
From Gus Wiseman, Oct 06 2023: (Start)
Also the number of subsets of {1..n} containing no two distinct elements summing to n. The a(0) = 1 through a(4) = 12 subsets are:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,3}  {4}
                  {2,3}  {1,2}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}
For n+1 instead of n we have A038754, complement A167762.
Including twins gives A117855, complement A366131.
The complement is counted by A365544.
For all subsets (not just pairs) we have A365377, complement A365376. (End)

			The a(3) = 6 walks: (0,-1,-2,-1), (0,-1,0,-1), (0,-1,0,1), (0,1,0,-1), (0,1,0,1), (0,1,2,1). - _Gus Wiseman_, Oct 10 2023

Alois P. Heinz, Table of n, a(n) for n = 0..4191
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
Robert Dorward et al., A Generalization of Zeckendorf's Theorem via Circumscribed m-gons, arXiv:1508.07531 [math.NT], 2015. See Example 1.3 p. 4.
Noam D. Elkies, New Directions in Enumerative Chess Problems, arXiv:math/0508645 [math.CO], 2005; The Electronic Journal of Combinatorics, 11 (2), 2004-2005.
Sean A. Irvine, Walks on Graphs.
D. Panario, M. Sahin, Q. Wang, and W. Webb, General conditional recurrences, Applied Mathematics and Computation, Volume 243, Sep 15 2014, Pages 220-231.
Noriaki Sannomiya, H. Katsura, and Y. Nakayama, Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion, arXiv preprint arXiv:1612.02285 [cond-mat.str-el], 2016-2017. See Table I, line 3.
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000007, A016116 (without initial term), A068912, A068913 for similar.
Equals A060647(n-1)+1.
Cf. also A028495, A038754, A048328, A078038, A124302, A306293.
First differences are A117855.
Cf. A004526, A004737, A008967, A046663, A088809, A365376, A365377, A365381, A365541, A365544, A366130.

[Floor((5-(-1)^n)*3^Floor(n/2)/3): n in [0..40]]; // Bruno Berselli, Feb 26 2016, after Charles R Greathouse IV

with(GraphTheory): G:= PathGraph(5): A:=AdjacencyMatrix(G): nmax:=34; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[3,k], k=1..5) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n+1, (4-irem(n, 2))/2*a(n-1)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 03 2019

Join[{1},Transpose[NestList[{Last[#],3First[#]}&,{2,4},40]][[1]]] (* Harvey P. Dale, Oct 24 2011 *)
Table[Length[Select[Subsets[Range[n]],FreeQ[Total/@Subsets[#,{2}],n]&]],{n,0,15}] (* Gus Wiseman, Oct 06 2023 *)

a(n)=[4,6][n%2+1]*3^(n\2)\3 \\ Charles R Greathouse IV, Feb 26 2016

def A068911(n): return 3**(n>>1)<<1 if n&1 else (3**(n-1>>1)<<2 if n else 1) # Chai Wah Wu, Aug 30 2024

a(n) = A068913(2, n) = 2*A038754(n-1) = 3*a(n-2) = a(n-1)*a(n-2)/a(n-3) starting with a(0)=1, a(1)=2, a(2)=4 and a(3)=6.
For n>0: a(2n) = 4*3^(n-1) = 2*a(2n-1); a(2n+1) = 2*3^n = 3*a(2n)/2 = 2*a(2n)-A000079(n-2).
From Paul Barry, Feb 17 2004: (Start)
G.f.: (1+x)^2/(1-3x^2).
a(n) = 2*3^((n+1)/2)*((1-(-1)^n)/6 + sqrt(3)*(1+(-1)^n)/9) - (1/3)*0^n.
The sequence 0, 1, 2, 4, ... has a(n) = 2*3^(n/2)*((1+(-1)^n)/6 + sqrt(3)*(1-(-1)^n)/9) - (2/3)*0^n + (1/3)*Sum_{k=0..n} binomial(n, k)*k*(-1)^k. (End)
a(n) = 2^((3 + (-1)^n)/2)*3^((2*n - 3 - (-1)^n)/4) - ((1 - (-1)^(2^n)))/6. - Luce ETIENNE, Aug 30 2014
For n > 2, indexing from 0, a(n) = a(n-1) + a(n-2) if n is odd, a(n-1) + a(n-2) + a(n-3) if n is even. - Alec Jones, Feb 25 2016
a(n) = 2*a(n-1) if n is even, a(n-1) + a(n-2) if n is odd. - Vincenzo Librandi, Feb 26 2016
E.g.f.: (4*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 1)/3. - Stefano Spezia, Feb 17 2022

A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

Original entry on oeis.org

1, 2, 3, 5, 10, 20, 40, 82, 169, 348, 716, 1471, 3016, 6171, 12605, 25710, 52370, 106539, 216470, 439310, 890550, 1803415, 3648557, 7375141, 14896184, 30065129, 60639954, 122231740, 246239551, 495790161, 997747182, 2006969629, 4035274292, 8110185100, 16293958314, 32724456982

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

			The a(0) = 1 through a(4) = 10 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {2,3}    {2,3}
                  {1,2,3}  {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

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
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A237668 counts sum-full partitions, ranks A364532.
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.
Triangles:
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A095944, A103580, A288728, A326080, A326083, A365376, A365544.

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

a(n) = 2^n - A367217(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 24, 46, 87, 164, 308, 577, 1080, 2021, 3779, 7058, 13166, 24533, 45674, 84978, 158026, 293737, 545747, 1013467, 1881032, 3489303, 6468910, 11985988, 22195905, 41080751, 75994642, 140514019, 259693004, 479749492, 885910870, 1635281386

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

			The a(2) = 1 through a(5) = 12 subsets:
  {2}  {2}    {2}    {2}
       {3}    {3}    {3}
       {1,3}  {4}    {4}
              {1,3}  {5}
              {1,4}  {1,3}
              {3,4}  {1,4}
                     {1,5}
                     {3,4}
                     {3,5}
                     {4,5}
                     {1,4,5}
                     {2,4,5}

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
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A229816 counts partitions whose length is not a part, complement A002865.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A237667 counts sum-free partitions, ranks A364531.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A103580, A240861, A288728, A326080, A326083, A364346, A364349, A365046, A365376, A365377.

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,15}]

a(n) = 2^n - A367216(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 23, 24, 29, 32, 37, 41, 49, 54, 63, 72, 82, 93, 108, 122, 139, 159, 180, 204, 231, 261, 293, 331, 370, 415, 464, 518, 575, 641, 710, 789, 871, 965, 1064, 1177, 1294, 1428, 1569, 1729, 1897

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367219.

			The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
  2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
                 43  53  54  64  65   75   76   86   87   97
                         63  73  74   84   85   95   96   A6
                                 83   93   94   A4   A5   B5
                                 542  642  A3   B3   B4   C4
                                           652  752  C3   D3
                                           742  842  654  754
                                                     762  862
                                                     852  952
                                                     942  A42

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.
A124506 appears to count combination-free subsets, differences of A326083.
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.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365541 counts subsets containing two distinct elements summing to k.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.

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,0,30}]

A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

1, 2, 3, 6, 12, 24, 49, 101, 207, 422, 859, 1747, 3548, 7194, 14565, 29452, 59496, 120086, 242185, 488035, 982672, 1977166, 3975508, 7989147, 16047464, 32221270, 64674453, 129775774, 260337978, 522124197, 1046911594, 2098709858, 4206361369, 8429033614, 16887728757, 33829251009, 67755866536, 135687781793, 271693909435

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			The set {1,2,4} has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(4).
The a(0) = 1 through a(4) = 12 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {1,3}    {1,3}
                  {2,3}    {1,4}
                  {1,2,3}  {2,3}
                           {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

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
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A326020 counts complete subsets.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts subsets containing two distinct elements summing to k.
Cf. A068911, A088314, A103580, A116861, A326080, A364350, A365073, A365311, A365376, A365380, A365544.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]], combs[Length[#], Union[#]]!={}&]], {n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367222(n):
    c, mlist = 1, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    c += 1
                    break
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367223(n).

a(13)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A366741 Number of semi-sums of strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 21, 26, 37, 48, 63, 86, 108, 139, 175, 223, 274, 350, 422, 527, 638, 783, 939, 1146, 1371, 1648, 1957, 2341, 2770, 3285, 3867, 4552, 5353, 6262, 7314, 8529, 9924, 11511, 13354, 15423, 17825, 20529, 23628, 27116, 31139, 35615

Offset: 0

Gus Wiseman, Nov 05 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 strict partitions of 9 and their a(9) = 13 semi-sums:
    (9) ->
   (81) -> 9
   (72) -> 9
   (63) -> 9
  (621) -> 3,7,8
   (54) -> 9
  (531) -> 4,6,8
  (432) -> 5,6,7

The non-strict non-binary version is A304792.
The non-binary version is A365925.
The non-strict version is A366738.
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.
A365543 counts partitions with a subset summing to k, complement A046663.
A365661 counts strict partitions w/ subset summing to k, complement A365663.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366739 counts semi-sums of prime indices, firsts A367097.
Cf. A008967, A122768, A299701, A364272, A364350, A365541, A365832, A366753.

Table[Total[Length[Union[Total/@Subsets[#, {2}]]]&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n,0,30}]

cp's OEIS Frontend

A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

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A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

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A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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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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A068911 Number of n-step walks (each step +-1 starting from 0) which are never more than 2 or less than -2.

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A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

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A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

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