A365830 -id:A365830 - cp's OEIS frontend

A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 0, 3, 3, 2, 1, 1, 0, 0, 0, 0, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 0, 0, 4, 8, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 2, 9, 9, 7, 5

Offset: 0

Gus Wiseman, Nov 09 2023

An integer partition of n is complete (ranks A325781) if every integer from 0 to n is the sum of some submultiset of the parts.

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  0  1  1
  0  0  0  2  1  1
  0  0  0  1  2  1  1
  0  0  0  1  3  2  1  1
  0  0  0  0  3  3  2  1  1
  0  0  0  0  4  5  3  2  1  1
  0  0  0  0  3  5  5  3  2  1  1
  0  0  0  0  4  8  7  5  3  2  1  1
  0  0  0  0  2  9  9  7  5  3  2  1  1
  0  0  0  0  2 11 12 11  7  5  3  2  1  1
  0  0  0  0  1 11 16 13 11  7  5  3  2  1  1
  0  0  0  0  1 14 21 19 15 11  7  5  3  2  1  1
Row n = 11 counts the following partitions (empty columns not shown):
  6311  62111  611111  5111111  41111111  311111111  2111111111  11111111111
  6221  53111  521111  4211111  32111111  221111111
  5321  52211  431111  3311111  22211111
  4421  44111  422111  3221111
        43211  332111  2222111
        42221  322211
        33311  222221
        33221

Column k appears to have A000325(k) nonzero terms.
Column sums are A003513.
Central column T(2n,n) is A007042.
Row sums are A126796, ranks A325781.
The strict case is too sparse, row sums A188431 (complement A365831).
Grouping by maximum instead of length gives A261036.
A000041 counts integer partitions.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts subset-sums of prime indices, firsts A259941.
A365924 counts incomplete partitions, ranks A365830.
Cf. A002033, A018818, A046663, A047967, A055932, A276024, A304792, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n,{k}],nmz[#]=={}&]],{n,0,15},{k,0,n}]

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

Offset: 0

Gus Wiseman, Nov 19 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

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A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

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