A365381 -id:A365381 - cp's OEIS frontend

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

0, 0, 1, 2, 7, 15, 38, 79, 184, 378, 823, 1682, 3552, 7208, 14948, 30154, 61698, 124302, 252125, 506521, 1022768, 2051555, 4127633, 8272147, 16607469, 33258510, 66680774, 133467385, 267349211, 535007304, 1071020315, 2142778192, 4288207796

Offset: 0

Gus Wiseman, Oct 07 2023

			The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
The a(0) = 0 through a(5) = 15 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,3}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {2,3,4}    {1,3,5}
                        {1,2,3,4}  {1,4,5}
                                   {2,3,4}
                                   {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}

For pairs summing to n + 1 we have A167762, complement A038754.
For n instead of n + 1 we have A365376, for pairs summing to n A365544.
The complement is counted by A365377 shifted.
The complement for pairs summing to n is counted by A365377.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A366130(n):
    a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
    return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # Chai Wah Wu, Nov 24 2023

Diagonal k = n + 1 of A365381.

a(20)-a(32) from Chai Wah Wu, Nov 24 2023

A365383 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53

Offset: 0

Gus Wiseman, Sep 08 2023

Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
  22, 15, 19, 17, 20, 17, 20, 16,
  22, 17, 20, 17, 21, 17, 20, 17,
  22, 17, 20, 17, 21, 17, 20, 17, ...

			Triangle begins:
   1
   2   1
   3   2   2
   5   3   4   3
   7   5   6   6   6
  11   7   9   8   9   7
  15  11  13  13  14  13  14
  22  15  19  17  20  17  20  16
  30  22  26  26  27  26  28  26  27
  42  30  37  34  39  33  40  34  39  34
  56  42  50  49  52  50  54  51  54  53  53
  77  56  68  64  71  63  73  63  71  65  70  62
 101  77  91  89  95  90  97  93  97  97  98  94  99
 135 101 122 115 127 115 130 114 131 119 130 117 132 116
 176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
  (6)       (51)      (51)      (51)      (51)      (51)
  (51)      (411)     (42)      (411)     (42)      (411)
  (42)      (321)     (411)     (33)      (411)     (321)
  (411)     (3111)    (321)     (321)     (321)     (3111)
  (33)      (2211)    (3111)    (3111)    (3111)    (2211)
  (321)     (21111)   (222)     (2211)    (222)     (21111)
  (3111)    (111111)  (2211)    (21111)   (2211)    (111111)
  (222)               (21111)   (111111)  (21111)
  (2211)              (111111)            (111111)
  (21111)
  (111111)

Column k = 0 is A000041, strict A000009.
The version for subsets is A365381, main diagonal A365376.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A088314, A088528, A237668, A363225, A364345, A365073, A365311, A365320, A365378, A365382.

combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]

cp's OEIS Frontend

A366130 Number of subsets of {1..n} with a subset summing to n + 1.

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