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

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

1,4

Links

Table of n, a(n) for n=1..78.

Formula

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

Example

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)

Mathematica

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

Crossrefs

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.

Sequence in context: A082860 A342859 A342385 * A283845 A365543 A058071

Adjacent sequences: A367105 A367106 A367107 * A367109 A367110 A367111

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 18 2023

Status

approved