A364272 -id:A364272 - cp's OEIS frontend

A116861 Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 0, 3, 2, 2, 2, 5, 1, 1, 3, 3, 2, 4, 2, 6, 1, 0, 5, 2, 3, 4, 4, 3, 8, 1, 1, 4, 3, 4, 7, 4, 5, 3, 10, 1, 0, 5, 3, 4, 7, 7, 6, 6, 5, 12, 1, 1, 6, 4, 3, 12, 6, 8, 7, 9, 5, 15, 1, 0, 6, 4, 5, 10, 10, 9, 10, 11, 10, 7, 18, 1, 1, 6, 4, 5, 15, 11, 13, 9, 16, 11, 13, 8, 22

Offset: 1

Emeric Deutsch, Feb 27 2006

Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019
Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)*A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to -A137719 without leading 0. - M. F. Hasler, Oct 27 2019
From Gus Wiseman, Aug 27 2023: (Start)
Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:
  7*1  .  1*2+5*1  1*3+4*1  1*3+2*2  1*5+2*1      1*7
          2*2+3*1  2*3+1*1  1*4+3*1  1*3+1*2+2*1  1*4+1*3
          3*2+1*1                                 1*5+1*2
                                                  1*6+1*1
                                                  1*4+1*2+1*1
(End)

			T(10,7) = 4 because we have [6,1,1,1,1], [4,3,3], [4,2,2,1,1] and [4,2,1,1,1,1] (6+1=4+3=4+2+1=7).
Triangle starts:
  1;
  1, 1;
  1, 0, 2;
  1, 1, 1, 2;
  1, 0, 2, 1, 3;
  1, 1, 3, 1, 1,  4;
  1, 0, 3, 2, 2,  2, 5;
  1, 1, 3, 3, 2,  4, 2, 6;
  1, 0, 5, 2, 3,  4, 4, 3, 8;
  1, 1, 4, 3, 4,  7, 4, 5, 3, 10;
  1, 0, 5, 3, 4,  7, 7, 6, 6,  5, 12;
  1, 1, 6, 4, 3, 12, 6, 8, 7,  9,  5, 15;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (16)(1).

Cf. A000041 (row sums), A000009 (diagonal), A014153.
Cf. A114638 (count partitions with #parts = sum(distinct parts)).
Column 1: A000012, column 2: A000035(1..), column 3: A137719(1..).
For subsets instead of partitions we have A026820.
This statistic is ranked by A066328.
The central diagonal is T(2n,n) = A364910(n), non-strict A364907.
Partial sums of columns are columns of A364911.
Same as A364916 (offset 0) with rows reversed.
A008284 counts partitions by length, strict A008289.
A364912 counts linear combinations of partitions.
A364913 counts combination-full partitions, strict A364839.
Cf. A237667, A364272, A364915, A365002, A365004, A365006.

g:= -1+product(1+t^j*x^j/(1-x^j), j=1..40): gser:= simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local f, g, j;
      if n=0 then [1] elif i<1 then [ ] else f:= b(n, i-1);
         for j to n/i do
           f:= zip((x, y)->x+y, f, [0$i, b(n-i*j, i-1)[]], 0)
         od; f
      fi
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 27 2013

max = 14; s = Series[-1+Product[1+t^j*x^j/(1-x^j), {j, 1, max}], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[Length[Select[IntegerPartitions[n],Total[Union[#]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Aug 29 2023 *)

A116861(n,k,s=0)={forpart(X=n,vecsum(Set(X))==k&&s++,k);s} \\ M. F. Hasler, Oct 27 2019

G.f.: -1 + Product_{j>=1} (1 + t^j*x^j/(1-x^j)).
Sum_{k=1..n} T(n,k) = A000041(n).
T(n,n) = A000009(n).
Sum_{k=1..n} k*T(n,k) = A014153(n-1).
T(n,1) = 1. T(n,2) = A000035(n+1). T(n,3) = A137719(n-2). - R. J. Mathar, Oct 27 2019
T(n,4) = A002264(n-1) + A121262(n). - R. J. Mathar, Oct 28 2019

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325

Offset: 0

Gus Wiseman, Aug 15 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(16) = 6 through a(22) = 12 strict partitions:
  (16)     (17)     (18)     (19)     (20)      (21)      (22)
  (9,7)    (9,8)    (10,8)   (10,9)   (11,9)    (12,9)    (13,9)
  (10,6)   (10,7)   (11,7)   (11,8)   (12,8)    (13,8)    (14,8)
  (11,5)   (11,6)   (13,5)   (12,7)   (13,7)    (15,6)    (15,7)
  (13,3)   (12,5)   (14,4)   (13,6)   (14,6)    (16,5)    (16,6)
  (7,5,4)  (13,4)   (7,6,5)  (14,5)   (17,3)    (17,4)    (17,5)
           (14,3)   (8,7,3)  (15,4)   (8,7,5)   (19,2)    (18,4)
           (15,2)            (16,3)   (9,6,5)   (11,10)   (19,3)
           (7,6,4)           (17,2)   (9,7,4)   (8,7,6)   (12,10)
                             (8,6,5)  (11,5,4)  (9,7,5)   (9,7,6)
                             (9,6,4)            (10,7,4)  (9,8,5)
                                                (10,8,3)  (7,6,5,4)
                                                (11,6,4)
                                                (11,7,3)

For sums of subsets instead of combinations of partitions we have A151897.
For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The complement in strict partitions is A364839, non-strict A364913.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A116861 and A364916 count linear combinations of strict partitions.
A323092 (ranks A320340) and A120641 count double-free partitions.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.

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@@#&&And@@Table[combs[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364350(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

More terms and offset corrected by Martin Fuller, Sep 11 2023

A088809 Number of subsets of {1, ..., n} that are not sum-free.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 27, 67, 154, 350, 763, 1638, 3450, 7191, 14831, 30398, 61891, 125557, 253841, 511818, 1029863, 2069341, 4153060, 8327646, 16687483, 33422562, 66916342, 133936603, 268026776, 536277032, 1072886163, 2146245056, 4293187682, 8587371116

Offset: 0

Reinhard Zumkeller, Oct 19 2003

nonn

a(n) = 2^n - A085489(n); a non-sum-free subset contains at least one subset {u,v, w} with w=u+v.

A variation of binary sum-full sets where parts cannot be re-used, this sequence counts subsets of {1..n} with an element equal to the sum of two distinct others. The complement is counted by A085489. The non-binary version is A364534. For re-usable parts we have A093971. - Gus Wiseman, Aug 10 2023

			From _Gus Wiseman_, Aug 10 2023: (Start)
The set S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are all missing from S, so it is not counted under a(8).
The set {1,4,6,7} has pair-sum 1 + 6 = 7, so is counted under a(7).
The a(1) = 0 through a(5) = 10 sets:
  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                 {1,3,4}    {1,3,4}
                 {1,2,3,4}  {1,4,5}
                            {2,3,5}
                            {1,2,3,4}
                            {1,2,3,5}
                            {1,2,4,5}
                            {1,3,4,5}
                            {2,3,4,5}
                            {1,2,3,4,5}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..75
Eric Weisstein's World of Mathematics, Sum-Free Set
Reinhard Zumkeller, Illustration of initial terms

The complement is counted by A085489, differences A364755.
With re-usable parts we have A093971, for partitions A363225.
The complement for partitions is A236912:
  non-binary A237667,
  ranks A364461,
  strict A364533.
The version for partitions is A237113:
  non-binary A237668,
  ranks A364462,
  strict A364670.
The non-binary version is A364534, complement A151897.
First differences are A364756.
Cf. A000079, A007865, A050291, A051026, A103580, A288728, A326020, A326080, A326083, A364272, A364349.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,10}] (* Gus Wiseman, Aug 10 2023 *)

Terms a(32) and beyond from Fausto A. C. Cariboni, Sep 28 2020

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

Original entry on oeis.org

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}]

A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 5, 3, 4, 4, 4, 4, 4, 4, 4, 7, 5, 7, 8, 7, 5, 7, 8, 7, 7, 8, 8, 7, 7, 8, 12, 9, 12, 9, 17, 9, 12, 9, 12, 14, 11, 12, 12, 13, 13, 12, 12, 11, 14, 21, 15, 19, 15, 21, 24, 21, 15, 19, 15, 21, 24, 19, 20, 19, 21, 22, 22, 21, 19, 20, 19, 24, 34, 23, 30, 24, 30, 25, 46, 25, 30, 24, 30, 23, 34

Offset: 2

N. J. A. Sloane

			For n = 4 there are two partitions (4, 2+2) with no subsum equal to 1, two (4, 3+1) with no subsum equal to 2 and two (4, 2+2) with no subsum equal to 3.
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  2,  2;
   2,  2,  2,  2;
   4,  3,  5,  3,  4;
   4,  4,  4,  4,  4,  4;
   7,  5,  7,  8,  7,  5,  7;
   8,  7,  7,  8,  8,  7,  7,  8;
  12,  9, 12,  9, 17,  9, 12,  9, 12;
  ...
From _Gus Wiseman_, Oct 11 2023: (Start)
Row n = 8 counts the following partitions:
  (8)     (8)    (8)     (8)     (8)     (8)    (8)
  (62)    (71)   (71)    (71)    (71)    (71)   (62)
  (53)    (53)   (62)    (62)    (62)    (53)   (53)
  (44)    (44)   (611)   (611)   (611)   (44)   (44)
  (422)   (431)  (44)    (53)    (44)    (431)  (422)
  (332)          (422)   (521)   (422)          (332)
  (2222)         (2222)  (5111)  (2222)         (2222)
                         (332)
(End)

Alois P. Heinz, Rows n = 2..98, flattened
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.

Column k = 0 and diagonal k = n are both A002865.
Central diagonal n = 2k is A006827.
The complement with expanded domain is A365543.
The strict case is A365663, complement A365661.
Row sums are A365918, complement A304792.
For subsets instead of partitions we have A366320, complement A365381.
A000041 counts integer partitions, strict A000009.
A276024 counts distinct subset-sums of partitions.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000124, A002219, A093971, A108917, A122768, A299701, A365658.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(0 in s or n in s, 0, `if`(n=0 or s={}, g(n, i),
     `if`(i<1, 0, b(n, i-1, s)+`if`(i>n, 0, b(n-i, i,
      select(y-> 0<=y and y<=n-i, map(x-> [x, x-i][], s)))))))
    end:
T:= (n, k)-> b(n, n, {min(k, n-k)}):
seq(seq(T(n, k), k=1..n-1), n=2..16);  # Alois P. Heinz, Jul 13 2012

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i > 1, g[n, i-1], 0] + If[i > n, 0, g[n-i, i]]]; b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0 || s == {}, g[n, i], If[i < 1, 0, b[n, i-1, s] + If[i > n, 0, b[n-i, i, Select[Flatten[s /. x_ :> {x, x-i}], 0 <= # <= n-i &]]]]]]; t[n_, k_] := b[n, n, {Min[k, n-k]}]; Table[t[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 20 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#],k]&]],{n,2,10},{k,1,n-1}] (* Gus Wiseman, Oct 11 2023 *)

Corrected and extended by Don Reble, Nov 04 2001

A237668 Number of partitions of n such that some part is a sum of two or more other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

			a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(0) = 0 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)
(End)

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

Cf. A179009.
The binary complement is A236912, ranks A364461.
The binary version is A237113, ranks A364462.
The complement is counted by A237667, ranks A364531.
The binary version with re-usable parts is A363225, ranks A364348.
The strict case is A364272.
The binary complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts distinct subset-sums of prime indices.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* Gus Wiseman, Aug 12 2023 *)

a(21)-a(47) from Giovanni Resta, Feb 22 2014

A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 27, 68, 156, 357, 775, 1667, 3505, 7303, 15019, 30759, 62489, 126619, 255542, 514721, 1034425, 2076924, 4164650, 8346306, 16715847, 33467324, 66982798, 134040148, 268179417, 536510608, 1073226084, 2146759579, 4293930436, 8588485846, 17177799658

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(0) = 0 through a(5) = 10 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,3,4}    {1,3,4}
                    {1,2,3,4}  {1,4,5}
                               {2,3,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Andrew Howroyd, Table of n, a(n) for n = 0..85

The binary version is A088809, complement A085489.
The complement is counted by A151897.
The complement for partitions is A237667, ranks A364531.
For partitions we have A237668, ranks A364532.
For strict partitions we have A364272, complement A364349.
A108917 counts knapsack partitions, strict A275972.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
Cf. A007865, A093971, A323092, A325862, A326083, A363225, A364345, A364346, A364348, A364350, A364533, A364670.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,10}]

a(n) = 2^n - A151897(n). - Andrew Howroyd, Jan 27 2024

a(16)-a(25) from Chai Wah Wu, Nov 14 2023

a(26) onwards (using A151897) added by Andrew Howroyd, Jan 27 2024

A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 9, 20, 48, 101, 219, 454, 944, 1917, 3925, 7915, 16004, 32188, 64751, 129822, 260489, 521672, 1045060, 2091808, 4187047, 8377255, 16762285, 33531228, 67077485, 134170217, 268371678, 536772231, 1073611321, 2147282291, 4294697258, 8589527163, 17179321094

Offset: 0

Gus Wiseman, Aug 17 2023

nonn

A variation of non-binary combination-full sets where parts can be re-used. The complement is counted by A326083. The binary version is A093971. For non-re-usable parts we have A364534. First differences are A365046.

			The set {3,4,5,17} has 17 = 1*3 + 1*4 + 2*5, so is counted under a(17).
The a(0) = 0 through a(5) = 20 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,3}    {1,3}      {1,3}
               {1,2,3}  {1,4}      {1,4}
                        {2,4}      {1,5}
                        {1,2,3}    {2,4}
                        {1,2,4}    {1,2,3}
                        {1,3,4}    {1,2,4}
                        {2,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,3,5}
                                   {1,4,5}
                                   {2,3,4}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865.
The binary version without re-usable parts is A088809.
The binary version is A093971.
The complement without re-usable parts is A151897.
The complement is counted by A326083.
The version without re-usable parts is A364534.
The version for strict partitions is A364839, complement A364350.
The version for partitions is A364913.
The version for positive combinations is A365043, complement A365044.
First differences are A365046.
Cf. A011782, A085489, A103580, A116861, A124506, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364756.

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]],Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A364914(n):
    c, mlist = 0, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m,k=m-1):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(2,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for d in w:
                for s in mlist[d-1]:
                    if s <= ws:
                        c += 1
                        break
                else:
                    continue
                break
    return c # Chai Wah Wu, Nov 17 2023

a(12)-a(34) from Chai Wah Wu, Nov 17 2023

A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 7, 12, 12, 17, 20, 26, 29, 39, 43, 54, 62, 77, 88, 107, 122, 148, 168, 200, 229, 267, 308, 360, 407, 476, 536, 623, 710, 812, 917, 1050, 1190, 1349, 1530, 1733, 1944, 2206, 2483, 2794, 3138, 3524

Offset: 0

Gus Wiseman, Aug 19 2023

nonn

			For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
  .  .  (21)  (31)  (41)  (42)   (61)   (62)   (63)   (82)    (A1)    (84)
                          (51)   (421)  (71)   (81)   (91)    (542)   (93)
                          (321)         (431)  (432)  (532)   (632)   (A2)
                                        (521)  (531)  (541)   (641)   (B1)
                                               (621)  (631)   (731)   (642)
                                                      (721)   (821)   (651)
                                                      (4321)  (5321)  (732)
                                                                      (741)
                                                                      (831)
                                                                      (921)
                                                                      (5421)
                                                                      (6321)

For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
Non-strict versions are A364913 and the complement of A364915.
For subsets instead of partitions we have A364914, complement A326083.
The case of no all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.

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@@#&&Or@@Table[combs[#[[k]], Delete[#,k]]!={}, {k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364839(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

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}]

cp's OEIS Frontend

A116861 Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).

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A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

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A088809 Number of subsets of {1, ..., n} that are not sum-free.

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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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A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

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A237668 Number of partitions of n such that some part is a sum of two or more other parts.

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A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

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A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.