A365663 -id:A365663 - cp's OEIS frontend

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Gus Wiseman, Sep 16 2023

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)

For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Cf. A000041, A008967, A095944, A364911, A365376, A365377.

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023

A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 25, 35, 53, 72, 108

Offset: 1

Gus Wiseman, Sep 30 2023

A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.

			The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
  {2}  {3}    {4}      {5}        {6}          {7}
       {1,3}  {1,4}    {1,5}      {1,6}        {1,7}
              {2,2}    {2,3}      {2,4}        {2,5}
              {1,1,4}  {1,1,5}    {3,3}        {3,4}
                       {1,2,5}    {1,1,6}      {1,1,7}
                       {1,1,1,5}  {1,2,6}      {1,2,7}
                                  {1,3,3}      {1,3,4}
                                  {2,2,2}      {2,2,3}
                                  {1,1,1,6}    {1,1,1,7}
                                  {1,1,2,6}    {1,1,2,7}
                                  {1,1,1,1,6}  {1,1,3,7}
                                               {1,2,2,7}
                                               {1,1,1,1,7}
                                               {1,1,1,2,7}
                                               {1,1,1,1,1,7}

For least instead of greatest we have A126796, ranks A325781, strict A188431.
These multisets have ranks A365830.
Counts appearances of n in the rank statistic A365920.
Column sums of A365921.
These multisets counted by sum are A365924, strict A365831.
The strict case is A366129.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365918 counts non-subset-sums of partitions.
A365923 counts partitions by non-subset sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365658, A365919, A365925.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

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

a(n) = binomial(A000009(n),2).

A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 7, 11, 11, 15, 18, 23, 28, 36, 40, 50, 59, 70, 83, 101, 118, 141, 166, 195, 227, 268, 306, 358, 414, 478, 549, 640, 730, 846, 968, 1113, 1271, 1462, 1657, 1897, 2154, 2451

Offset: 1

Gus Wiseman, Oct 07 2023

A non-subset-sum of a set summing to n is a positive integer up to n that is not the sum of any subset. For example, the non-subset-sums of {1,3,4} are {2,6}.

			The a(1) = 1 through a(8) = 11 sets:
  {2}  {3}    {4}    {5}      {6}      {7}      {8}        {9}
       {1,3}  {1,4}  {2,3}    {2,4}    {2,5}    {2,6}      {2,7}
                     {1,5}    {1,6}    {3,4}    {3,5}      {3,6}
                     {1,2,5}  {1,2,6}  {1,7}    {1,8}      {4,5}
                                       {1,3,4}  {1,3,5}    {2,3,4}
                                       {1,2,7}  {1,2,8}    {1,9}
                                                {1,2,3,8}  {1,3,6}
                                                           {1,4,5}
                                                           {1,2,9}
                                                           {1,2,3,9}
                                                           {1,2,4,9}

For least instead of greatest: A188431, non-strict A126796 (ranks A325781).
The version counting multisets instead of sets is A366127.
These sets counted by sum are A365924, strict A365831.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A365923 counts partitions by number of non-subset-sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n], UnsameQ@@#&&Max@@nmz[#]==n&]],{n,15}]

a(31)-a(42) from Erich Friedman, Nov 13 2024

A367108 Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 2, 3, 5, 7, 5, 4, 4, 5, 7, 11, 7, 6, 3, 6, 7, 11, 15, 11, 8, 7, 7, 8, 11, 15, 22, 15, 12, 10, 4, 10, 12, 15, 22, 30, 22, 16, 14, 12, 12, 14, 16, 22, 30, 42, 30, 22, 17, 17, 6, 17, 17, 22, 30, 42, 56, 42, 30, 25, 23, 20, 20, 23, 25, 30, 42, 56

Offset: 1

Gus Wiseman, Nov 18 2023

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   2   3   5
   7   5   4   4   5   7
  11   7   6   3   6   7  11
  15  11   8   7   7   8  11  15
  22  15  12  10   4  10  12  15  22
  30  22  16  14  12  12  14  16  22  30
  42  30  22  17  17   6  17  17  22  30  42
  56  42  30  25  23  20  20  23  25  30  42  56
  77  56  40  31  30  27   7  27  30  31  40  56  77
Row n = 5 counts the following partitions:
  (5)      (41)     (32)     (32)     (41)     (5)
  (41)     (311)    (311)    (311)    (311)    (41)
  (32)     (221)    (221)    (221)    (221)    (32)
  (311)    (2111)   (11111)  (11111)  (2111)   (311)
  (221)    (11111)                    (11111)  (221)
  (2111)                                       (2111)
  (11111)                                      (11111)
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (2211)    (411)     (411)     (51)
  (42)      (321)     (321)     (111111)  (321)     (321)     (42)
  (411)     (3111)    (3111)              (3111)    (3111)    (411)
  (33)      (2211)    (222)               (222)     (2211)    (33)
  (321)     (21111)   (111111)            (111111)  (21111)   (321)
  (3111)    (111111)                                (111111)  (3111)
  (222)                                                       (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Columns k = 0 and k = n are A000041(n).
Column k = 1 and k = n-1 are A000041(n-1).
Column k = 2 appears to be 2*A027336(n).
The version for non-subset-sums is A046663, strict A365663.
Diagonal n = 2k is A108917, complement A366754.
Row sums are A304796, non-unique version A304792.
The non-unique version is A365543.
Cf. A002219, A122768, A275972, A299702, A299729, A301854, A364272, A364911, A365658, A365661, A367094.

Table[Length[Select[IntegerPartitions[n], Count[Total/@Union[Subsets[#]], k]==1&]], {n,0,10}, {k,0,n}]

A367094(n,1) = A108917(n).

cp's OEIS Frontend

A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

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A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

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A366132 Number of unordered pairs of distinct strict integer partitions of n.

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A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

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A367108 Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.

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