A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 23, 24, 29, 32, 37, 41, 49, 54, 63, 72, 82, 93, 108, 122, 139, 159, 180, 204, 231, 261, 293, 331, 370, 415, 464, 518, 575, 641, 710, 789, 871, 965, 1064, 1177, 1294, 1428, 1569, 1729, 1897

Offset

0,8

Comments

The non-strict version is A367219.

Links

Table of n, a(n) for n=0..59.

Example

The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
  2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
                 43  53  54  64  65   75   76   86   87   97
                         63  73  74   84   85   95   96   A6
                                 83   93   94   A4   A5   B5
                                 542  642  A3   B3   B4   C4
                                           652  752  C3   D3
                                           742  842  654  754
                                                     762  862
                                                     852  952
                                                     942  A42

Mathematica

combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]=={}&]], {n, 0, 30}]

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.

sum-full sum-free comb-full comb-free

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partitions: A367212 A367213 A367218 A367219

strict: A367214 A367215 A367220 A367221*

subsets: A367216 A367217 A367222 A367223

ranks: A367224 A367225 A367226 A367227

A000041 counts integer partitions, strict A000009.

A002865 counts partitions whose length is a part, complement A229816.

A124506 appears to count combination-free subsets, differences of A326083.

A188431 counts complete strict partitions, incomplete A365831.

A240855 counts strict partitions whose length is a part, complement A240861.

A364272 counts sum-full strict partitions, sum-free A364349.

Triangles:

A008284 counts partitions by length, strict A008289.

A046663 counts partitions of n without a subset-sum k, strict A365663.

A365541 counts subsets containing two distinct elements summing to k.

A365658 counts partitions by number of subset-sums, strict A365832.

Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.

Sequence in context: A029019 A040039 A008667 * A239880 A240862 A177716

Adjacent sequences: A367218 A367219 A367220 * A367222 A367223 A367224

Keyword

nonn

Author

Gus Wiseman, Nov 14 2023

Status

approved