A196723 -id:A196723 - cp's OEIS frontend

A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 46, 58, 77, 93, 122, 146, 188, 217, 282, 327, 410, 470, 596, 673, 848, 947, 1178, 1325, 1629, 1798, 2213, 2444, 2962, 3247, 3935, 4292, 5149, 5579, 6674, 7247, 8590, 9221, 10964, 11804, 13870, 14843, 17480, 18675, 21866

Offset: 0

Gus Wiseman, May 31 2019

nonn

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The three non-knapsack partitions counted under a(6) are:
  (2,2,1,1)
  (3,1,1,1)
  (2,1,1,1,1)

Dominates A108917.

Cf. A002033, A034444, A196723, A275972, A276024, A299702, A325592, A325856, A325863, A325864, A325865, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#]]&]],{n,0,20}]

A196724 Number of subsets of {1..n} (including empty set) such that the pairwise products of distinct elements are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 58, 116, 212, 416, 720, 1440, 2340, 4680, 7920, 13024, 23328, 46656, 74168, 148336, 229856, 371424, 615304, 1230608, 1780224, 3401568, 5589360, 9468504, 14397744, 28795488, 40312128, 80624256, 131388480, 206363168, 335814288, 521401536

Offset: 0

Alois P. Heinz, Oct 06 2011

nonn

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product is A325860(n). - Gus Wiseman, Jun 03 2019

			a(6) = 58: from the 2^6=64 subsets of {1,2,3,4,5,6} only 6 do not have all the pairwise products of elements distinct: {1,2,3,6}, {2,3,4,6}, {1,2,3,4,6}, {1,2,3,5,6}, {2,3,4,5,6}, {1,2,3,4,5,6}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..50

Cf. A143823, A196719, A196720, A196721, A196722, A196723.
The subset case is A196724 (this sequence).
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Heinz numbers of the counterexamples are given by A325993.
Cf. A292886, A293627, A325860, A325861.

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       sn[i]*sn[j], j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] * sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]]; a[n_] := a[n] = b[n - 1, {n}] + If[n == 0, 0, a[n - 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Times@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Name edited by Gus Wiseman, Jun 03 2019

a(33)-a(35) from Fausto A. C. Cariboni, Oct 05 2020

A325860 Number of subsets of {1..n} such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 8, 14, 28, 52, 104, 188, 308, 548, 1096, 1784, 3568, 6168, 10404, 16200, 32400, 49968, 99936, 155584, 256944, 433736, 867472, 1297504, 2026288, 3387216, 5692056, 8682912, 17365824, 25243200, 50486400, 78433056, 125191968, 206649216, 328195632

Offset: 0

Gus Wiseman, May 31 2019

nonn

Also subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}    {}     {}
      {1}  {1}   {1}    {1}
           {2}   {2}    {2}
           {12}  {3}    {3}
                 {12}   {4}
                 {13}   {12}
                 {23}   {13}
                 {123}  {14}
                        {23}
                        {24}
                        {34}
                        {123}
                        {134}
                        {234}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..50

The subset case is A325860.
The maximal case is A325861.
The integer partition case is A325853.
The strict integer partition case is A325854.
Heinz numbers of the counterexamples are given by A325994.
Cf. A002033, A108917, A143823, A196723, A196723, A196724, A325855, A325858, A325859, A325868, A325869.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,0,20}]

a(21)-a(25) from Alois P. Heinz, Jun 07 2019

a(26)-a(35) from Fausto A. C. Cariboni, Oct 04 2020

A325864 Number of subsets of {1..n} of which every subset has a different sum.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 56, 91, 135, 211, 307, 446, 625, 882, 1194, 1677, 2238, 3031, 4001, 5460, 6995, 9302, 11921, 15424, 19554, 25032, 31005, 39170, 48251, 59917, 73093, 90831, 109271, 134049, 160922, 196109, 234179, 284157, 335933, 408390, 482597, 575109

Offset: 0

Gus Wiseman, Jun 01 2019

nonn

			The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..150 (terms 0..60 from Giovanni Resta)

Cf. A002033, A108917, A143823, A196723, A275972.

Cf. A325860, A325865, A325866, A325867, A325877, A325878, A325879, A325880.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#]&]],{n,0,10}]

a(18)-a(42) from Alois P. Heinz, Jun 03 2019

A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 18, 19, 26, 28, 36, 37, 50, 52, 67, 68, 89, 94, 115, 121, 151, 160, 195, 200, 247, 265, 312, 329, 386, 418, 487, 519, 600, 640, 742, 792, 901, 978, 1088, 1185, 1331, 1453, 1605, 1729, 1925, 2101, 2311, 2524, 2741, 3000

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

The non-strict case is A325857.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)   (43)   (53)   (54)   (64)
                        (41)  (51)   (52)   (62)   (63)   (73)
                              (321)  (61)   (71)   (72)   (82)
                                     (421)  (431)  (81)   (91)
                                            (521)  (432)  (532)
                                                   (531)  (541)
                                                   (621)  (631)
                                                          (721)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450

The subset case is A196723.
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A275972.
Cf. A325854, A325855, A325858, A325862, A325864, A325876, A325879.

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

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 8, 22, 40, 56, 78, 124, 222, 390, 616, 892, 1220, 1620, 2182, 3042, 4392, 6364, 9054, 12608, 16980, 22244, 28482, 36208, 45864, 58692, 75804, 98440, 128694, 168250, 218558, 281210, 357594, 449402, 560034, 693332, 853546, 1050118, 1293458, 1596144, 1975394

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

			The a(1) = 1 through a(6) = 8 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,3}  {1,2,4}    {1,2,3,5}
                       {1,2,4}  {2,3,4}    {1,2,3,6}
                       {1,3,4}  {2,4,5}    {1,2,4,6}
                       {2,3,4}  {1,2,3,5}  {1,3,4,5}
                                {1,3,4,5}  {1,3,5,6}
                                           {1,4,5,6}
                                           {2,3,4,6}
                                           {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..60

The subset case is A196723.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A143824, A276024.
Cf. A325858, A325859, A325864, A325865, A325867, A325879, A325880, A382398.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,b< n, ismaxl(b,w),
         my(s=self()(k+1, r, b, w));
         if(!bitand(w,b<Andrew Howroyd, Mar 23 2025

a(21) onwards from Andrew Howroyd, Mar 23 2025

A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 20, 24, 36, 64, 110, 176, 238, 294, 370, 504, 736, 1086, 1592, 2240, 2982, 3788, 4700, 5814, 7322, 9396, 12336, 16552, 22192, 29310, 38046, 48368, 60078, 73722, 89416, 108208, 131310, 160624, 198002, 247408, 310410, 390924, 490818, 613344, 758518

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

Also the number of maximal subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum.

			The a(0) = 1 through a(7) = 20 subsets:
  {}  {1}  {1,2}  {1,2}  {2,3}    {1,2,4}  {1,2,4}  {1,2,4}
                  {1,3}  {1,2,4}  {1,2,5}  {1,2,5}  {1,2,6}
                  {2,3}  {1,3,4}  {1,3,4}  {1,2,6}  {1,3,4}
                                  {1,4,5}  {1,3,4}  {1,4,5}
                                  {2,3,5}  {1,3,6}  {1,4,6}
                                  {2,4,5}  {1,4,5}  {1,5,6}
                                           {1,4,6}  {2,3,5}
                                           {1,5,6}  {2,3,6}
                                           {2,3,5}  {2,3,7}
                                           {2,3,6}  {2,4,5}
                                           {2,4,5}  {2,4,7}
                                           {2,5,6}  {2,5,6}
                                           {3,4,6}  {2,6,7}
                                           {3,5,6}  {3,4,6}
                                                    {3,4,7}
                                                    {3,5,6}
                                                    {4,5,7}
                                                    {4,6,7}
                                                    {1,2,5,7}
                                                    {1,3,6,7}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

The subset case is A143823.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A108917, A143824, A196723, A382397.
Cf. A325859, A325861, A325865, A325867, A325869, A325878, A325992.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1< n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 27 2025

a(21)-a(45) from Fausto A. C. Cariboni, Feb 08 2022

A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125

Offset: 0

Gus Wiseman, Jul 31 2023

nonn

In other words, the elements are not disjoint from their own first differences.

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,2,3}  {2,4}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,4,5}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {1,2,3,4,5}

For differences of all pairs we have A093971, complement A196723.
For partitions we have A363260, complement A364467.
The complement is counted by A364463.
For subset-sums instead of differences we have A364534, complement A325864.
For strict partitions we have A364536, complement A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A108917 counts knapsack partitions, strict A275972.
A325325 counts partitions with all distinct differences, strict A320347.
Cf. A011782, A212986, A237113, A325877, A325878, A326083, A363225.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]

from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) = 2^n - A364463(n). - Chai Wah Wu, Sep 26 2023

a(21)-a(32) from Chai Wah Wu, Sep 26 2023

a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 13, 21, 28, 42, 55, 78, 106, 144, 187, 255, 325, 429, 554, 717, 906, 1165, 1460, 1853, 2308, 2899, 3582, 4468, 5489, 6779, 8291, 10173, 12363, 15079, 18247, 22124, 26645, 32147, 38555, 46285, 55310, 66093, 78684, 93674, 111104

Offset: 0

Gus Wiseman, Jul 31 2023

nonn

In other words, the parts are not disjoint from their own first differences.

			The a(3) = 1 through a(9) = 13 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (621)
                       (2211)   (3211)    (521)      (3321)
                       (21111)  (22111)   (3221)     (4221)
                                (211111)  (4211)     (4311)
                                          (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

For all differences of pairs parts we have A363225, complement A364345.
The complement is counted by A363260.
For subsets of {1..n} we have A364466, complement A364463.
The strict case is A364536, complement A364464.
These partitions have ranks A364537.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]!={}&]],{n,0,30}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A364467(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 03 2019

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

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A196724 Number of subsets of {1..n} (including empty set) such that the pairwise products of distinct elements are all distinct.

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A325860 Number of subsets of {1..n} such that every pair of distinct elements has a different quotient.

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A325864 Number of subsets of {1..n} of which every subset has a different sum.

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A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

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A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

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A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.

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A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

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