A364916 -id:A364916 - 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

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}]

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}]

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

A088314 Cardinality of set of sets of parts of all partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 51, 61, 79, 96, 124, 148, 186, 222, 275, 326, 400, 473, 575, 673, 811, 946, 1132, 1317, 1558, 1813, 2138, 2463, 2893, 3323, 3882, 4461, 5177, 5917, 6847, 7818, 8994, 10251, 11766, 13334, 15281, 17309, 19732, 22307

Offset: 0

Naohiro Nomoto, Nov 05 2003

Number of different values of A007947(m) when A056239(m) is equal to n.
From Gus Wiseman, Sep 11 2023: (Start)
Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:
  {1}  {1}  {1}    {1}    {1}    {1}      {1}
       {2}  {3}    {2}    {5}    {2}      {7}
            {1,2}  {4}    {1,2}  {3}      {1,2}
                   {1,2}  {1,3}  {6}      {1,3}
                   {1,3}  {1,4}  {1,2}    {1,4}
                          {2,3}  {1,3}    {1,5}
                                 {1,4}    {1,6}
                                 {1,5}    {2,3}
                                 {2,4}    {2,5}
                                 {1,2,3}  {3,4}
                                          {1,2,3}
                                          {1,2,4}
Cf. A365073, A365311, A365312, A365322, A365380.
(End)

			The 7 partitions of 5 and their sets of parts are
[ #]  partition      set of parts
[ 1]  [ 1 1 1 1 1 ]  {1}
[ 2]  [ 2 1 1 1 ]    {1, 2}
[ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
[ 4]  [ 3 1 1 ]      {1, 3}
[ 5]  [ 3 2 ]        {2, 3}
[ 6]  [ 4 1 ]        {1, 4}
[ 7]  [ 5 ]          {5}
so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A182410.
The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.
The case of pairs is A365315, see also A365314, A365320, A365321.
A116861 and A364916 count linear combinations of strict partitions.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364914 and A365046 count combination-full subsets.
Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

list2set := L -> {op(L)};
a:= N -> list2set(map( list2set, combinat[partition](N) ));
seq(nops(a(n)), n=0..30);
#  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[],i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);
# Alois P. Heinz, Aug 09 2012

Table[Length[Union[Map[Union,IntegerPartitions[n]]]],{n,1,30}] (* Geoffrey Critzer, Feb 19 2013 *)
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
     Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
     Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
a[n_] := Length[b[n, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
combp[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,1,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&&combp[n,#]!={}&]], {n,0,15}] (* Gus Wiseman, Sep 11 2023 *)

from sympy.utilities.iterables import partitions
def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

a(n) = 2^(n-1) - A070880(n). - Alois P. Heinz, Feb 08 2019

a(n) = A365042(n) + 1. - Gus Wiseman, Sep 13 2023

More terms and clearer definition from Vladeta Jovovic, Apr 21 2005

A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 30, 34, 50, 58, 79, 96, 129, 152, 203, 243, 307, 375, 474, 563, 707, 850, 1042, 1246, 1532, 1815, 2215, 2632, 3173, 3765, 4525, 5323, 6375, 7519, 8916, 10478, 12414, 14523, 17133, 20034, 23488, 27422, 32090, 37285, 43511, 50559

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - Gus Wiseman, Apr 21 2025

			The a(1) = 1 through a(9) = 16 partitions:
 (1) (2) (3)  (4)   (5)   (6)   (7)    (8)    (9)
         (21) (31)  (32)  (42)  (43)   (53)   (54)
              (211) (41)  (51)  (52)   (62)   (63)
                    (221) (321) (61)   (71)   (72)
                    (311) (411) (322)  (332)  (81)
                                (331)  (422)  (432)
                                (421)  (431)  (441)
                                (511)  (521)  (522)
                                (3211) (611)  (531)
                                       (3221) (621)
                                       (4211) (711)
                                              (3321)
                                              (4221)
                                              (4311)
                                              (5211)
                                              (32211)

Chai Wah Wu, Table of n, a(n) for n = 0..160

The case with parts also covering an initial interval is A317088.
These partitions are ranked by A317090.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047966 counts partitions with constant section-sums.
A048767 interchanges prime indices and prime multiplicities (Look-and-Say), see A048768.
A055932 lists numbers whose prime indices cover an initial interval.
A116540 counts normal set multipartitions.
A304442 counts partitions with equal run-sums, ranks A353833.
A381436 lists the section-sum partition of prime indices.
A381440 lists the Look-and-Say partition of prime indices.
Cf. A000837, A069799, A217605, A317082, A317084, A317085, A317087.
Cf. A003242, A116861, A239455, A353837, A364916, A381431, A381432, A381438.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],normalQ[Length/@Split[#]]&]],{n,30}]

from sympy.utilities.iterables import partitions
def A317081(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n):
        s = set(d.values())
        if len(s) == max(s):
            c += 1
    return c # Chai Wah Wu, Jun 22 2020

A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 12, 20, 27, 39, 51, 74, 95, 130, 169, 225, 288, 378, 479, 617, 778, 990, 1239, 1560, 1938, 2419, 2986, 3696, 4538, 5575, 6810, 8319, 10102, 12274, 14834, 17932, 21587, 25963, 31120, 37275, 44513, 53097, 63181, 75092, 89030, 105460, 124647

Offset: 0

Gus Wiseman, Aug 20 2023

nonn

Includes all non-strict partitions (A047967).

			The a(0) = 0 through a(7) = 12 partitions:
  .  .  (11)  (21)   (22)    (41)     (33)      (61)
              (111)  (31)    (221)    (42)      (322)
                     (211)   (311)    (51)      (331)
                     (1111)  (2111)   (222)     (421)
                             (11111)  (321)     (511)
                                      (411)     (2221)
                                      (2211)    (3211)
                                      (3111)    (4111)
                                      (21111)   (22111)
                                      (111111)  (31111)
                                                (211111)
                                                (1111111)
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(12).
The partition (6,4,3,2) has 6 = 4+2, or 6 = 3+3, or 6 = 2+2+2, or 4 = 2+2, so is counted under a(15).

The strict case is A364839.
For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
For subsets instead of partitions we have A364914, complement A326083.
Allowing equal parts gives A365068, complement A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A365006 = no strict partitions w/ pos linear combination.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A364346.

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}]

a(n) + A364915(n) = A000041(n).

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

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

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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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A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

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A088314 Cardinality of set of sets of parts of all partitions of n.

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A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.