A275972 -id:A275972 - cp's OEIS frontend

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 8, 9, 12, 13, 16, 19, 21, 25, 30, 36, 40, 47, 53, 63, 71, 83, 94, 107, 121, 140, 159, 180, 204, 233, 260, 296, 334, 377, 421, 474, 532, 598, 668, 750, 835, 933, 1038, 1163, 1292, 1435, 1597, 1771, 1966, 2180, 2421, 2673

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(13) = 8 strict integer partitions (A = 10, B = 11, C = 12, D = 13):
  (2)  (3)  (4)  (5)   (6)  (7)   (8)   (9)   (A)    (B)    (C)    (D)
                 (32)       (43)  (53)  (54)  (64)   (65)   (75)   (76)
                            (52)  (62)  (63)  (73)   (74)   (84)   (85)
                                        (72)  (532)  (83)   (A2)   (94)
                                                     (92)   (543)  (A3)
                                                     (632)  (732)  (B2)
                                                                   (643)
                                                                   (652)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

Cf. A001597, A007916, A025147, A052410, A078374, A087897, A120641, A275972, A303362, A323053, A323087, A323088.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],UnsameQ@@#,stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995

Offset: 1

Gus Wiseman, Jun 03 2019

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

These are the subsets counted by A325867, ordered by sum rather than maximum.

			The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):
  {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}
                                              {5,3,1}  {7,2,1}  {6,3,2}
                                              {6,2,1}           {6,4,1}
                                                                {7,3,1}
.
  {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}
  {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}
  {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}
  {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}
           {9,3,1}                                 {9,5,2,1}
                                                   {10,4,2,1}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..600

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]
maxsks[n_]:=Select[Select[IntegerPartitions[n],UnsameQ@@#&&ksQ[#]&],Select[Table[Append[#,i],{i,Complement[Range[Max@@#],#]}],ksQ]=={}&];
Table[Length[maxsks[n]],{n,30}]

A326034 Number of knapsack partitions of n with largest part 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.
Appears to repeat the terms (2,2,2,3,1,3) ad infinitum.
I computed terms a(n) for n = 0..5000 and (2,2,2,3,1,3) is repeated continuously starting at a(8). - Fausto A. C. Cariboni, May 14 2021

			The initial values count the following partitions:
   3: (3)
   4: (3,1)
   5: (3,2)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   7: (3,2,2)
   8: (3,3,2)
   8: (3,3,1,1)
   9: (3,3,3)
   9: (3,2,2,2)
  10: (3,3,3,1)
  10: (3,3,2,2)
  11: (3,3,3,2)
  11: (3,3,3,1,1)
  11: (3,2,2,2,2)
  12: (3,3,3,3)
  13: (3,3,3,3,1)
  13: (3,3,3,2,2)
  13: (3,2,2,2,2,2)
  14: (3,3,3,3,2)
  14: (3,3,3,3,1,1)
  15: (3,3,3,3,3)
  15: (3,2,2,2,2,2,2)

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325592, A326015, A326016, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
kst[n_]:=Select[IntegerPartitions[n,All,{1,2,3}],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[kst[n],Max@@#==3&]],{n,0,30}]

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

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

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 5, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 4, 3, 3, 4, 3, 3, 8, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 7, 6, 7, 7, 7, 4, 7, 7, 7, 6, 7, 15, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Clark Kimberling, Feb 05 2014

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000

Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 3, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 3, 3, 1, 2, 1, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 5

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

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

Number of distinct values obtained when A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[LCM@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 4, 6, 8, 8

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of product-sum knapsack partitions begins:
   0: ()
   1:
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3) (4,4,3)
  12: (12) (10,2) (9,3) (8,4) (7,5) (7,3,2) (6,6) (4,4,4)
A complete list of all products of sums of multiset partitions of submultisets of (4,3,3) is:
           () = 1
          (3) = 3
          (4) = 4
        (3+3) = 6
        (3+4) = 7
      (3+3+4) = 10
      (3)*(3) = 9
      (3)*(4) = 12
    (3)*(3+4) = 21
    (4)*(3+3) = 24
  (3)*(3)*(4) = 36
These are all distinct, so (4,3,3) is a product-sum knapsack partition of 10.

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320053, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rrsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,Union@@mps/@Union[Subsets[q]],{2}],{1}]]];
Table[Length[rrsuks[n]],{n,12}]

A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 8, 9, 12, 14

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of spanning sum-product knapsack partitions begins:
  0: ()
  1: (1)
  2: (2) (1,1)
  3: (3) (2,1) (1,1,1)
  4: (4) (3,1)
  5: (5) (4,1) (3,2)
  6: (6) (5,1) (4,2) (3,3)
  7: (7) (6,1) (5,2) (4,3) (3,3,1)
  8: (8) (7,1) (6,2) (5,3) (4,4) (3,3,2)
  9: (9) (8,1) (7,2) (6,3) (5,4) (4,4,1) (4,3,2) (3,3,3)
A complete list of all sums of products covering the parts of (3,3,3,2) is:
        (2*3*3*3) = 54
      (2)+(3*3*3) = 29
      (3)+(2*3*3) = 21
      (2*3)+(3*3) = 15
    (2)+(3)+(3*3) = 14
    (3)+(3)+(2*3) = 12
  (2)+(3)+(3)+(3) = 11
These are all distinct, so (3,3,3,2) is a spanning sum-product knapsack partition of 11.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,mps[q],{2}],{1}]]];
Table[Length[rtuks[n]],{n,8}]

A320054 Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 5, 8, 10, 12, 16, 17, 25

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of spanning product-sum knapsack partitions begins
0: ()
1: (1)
2: (2) (1,1)
3: (3) (2,1) (1,1,1)
4: (4) (3,1)
5: (5) (4,1) (3,2) (3,1,1)
6: (6) (5,1) (4,2) (4,1,1) (3,3)
7: (7) (6,1) (5,2) (5,1,1) (4,3) (4,2,1) (4,1,1,1) (3,3,1)
8: (8) (7,1) (6,2) (6,1,1) (5,3) (5,2,1) (5,1,1,1) (4,4) (4,3,1) (3,3,2)
9: (9) (8,1) (7,2) (7,1,1) (6,3) (6,2,1) (6,1,1,1) (5,4) (5,3,1) (4,4,1) (4,3,2) (3,3,3)
A complete list of all products of sums covering the parts of (4,1,1,1) is:
        (1+1+1+4) = 7
      (1)*(1+1+4) = 6
      (4)*(1+1+1) = 12
      (1+1)*(1+4) = 10
    (1)*(1)*(1+4) = 5
    (1)*(4)*(1+1) = 8
  (1)*(1)*(1)*(4) = 4
These are all distinct, so (4,1,1,1) is a spanning product-sum knapsack partition of 7.
A complete list of all products of sums covering the parts of (5,3,1,1) is:
        (1+1+3+5) = 10
      (1)*(1+3+5) = 9
      (3)*(1+1+5) = 21
      (5)*(1+1+3) = 25
      (1+1)*(3+5) = 16
      (1+3)*(1+5) = 24
    (1)*(1)*(3+5) = 8
    (1)*(3)*(1+5) = 18
    (1)*(5)*(1+3) = 20
    (3)*(5)*(1+1) = 30
  (1)*(1)*(3)*(5) = 15
These are all distinct, so (5,3,1,1) is a spanning product-sum knapsack partition of 10.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320052, A320053, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,mps[q],{2}],{1}]]];
Table[Length[rsuks[n]],{n,10}]

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 6, 8, 18, 16, 30, 34, 48, 48, 72, 72, 96, 98, 126, 128, 162, 160, 198, 202, 240, 240, 288, 288, 336, 338, 390, 392, 450, 448, 510, 514, 576, 576, 648, 648, 720, 722, 798, 800, 882, 880, 966, 970, 1056, 1056, 1152, 1152, 1248, 1250, 1350, 1352

Offset: 0

Gus Wiseman, May 13 2019

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
From Kevin O'Bryant, Jun 02 2025: (Start)
Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.
Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)

			The a(6) = 2 through a(10) = 16 compositions:
  (132)  (124)  (125)  (126)  (127)
  (231)  (142)  (143)  (135)  (136)
         (214)  (152)  (153)  (154)
         (241)  (215)  (162)  (163)
         (412)  (251)  (216)  (172)
         (421)  (341)  (234)  (217)
                (512)  (243)  (253)
                (521)  (261)  (271)
                       (315)  (316)
                       (324)  (352)
                       (342)  (361)
                       (351)  (451)
                       (423)  (613)
                       (432)  (631)
                       (513)  (712)
                       (531)  (721)
                       (612)
                       (621)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Column k = 3 of A325677.
Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.
Cf. A325676, A325688, A325689, A325690, A325691.

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]

Conjectures from Colin Barker, May 14 2019: (Start)
G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9. (End)
Above conjecture confirmed for n <= 5000. - Fausto A. C. Cariboni, Feb 17 2022

cp's OEIS Frontend

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326034 Number of knapsack partitions of n with largest part 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A320054 Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.

Views

Author

Keywords

Examples