A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset

0,7

Links

Table of n, a(n) for n=0..77.

S. R. Finch, Monoids of natural numbers, March 17, 2009.

Example

The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

Mathematica

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], Max@@Prepend[nmz[#], 0]==k&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

Row sums are A000041.

Diagonal k = n-1 is A002865.

Column k = 1 is A126796 (complete partitions), ranks A325781.

Central diagonal n = 2k is A126796 also.

For parts instead of sums we have A339737, rank stat A339662, min A257993.

This is the triangle for the rank statistic A365920.

Latter row sums are A365924 (incomplete partitions), ranks A365830.

Column sums are A366127.

A055932 lists numbers whose prime indices cover an initial interval.

A056239 adds up prime indices, row sums of A112798.

A073491 lists numbers with gap-free prime indices.

A238709/A238710 count partitions by least/greatest difference.

A342050/A342051 have prime indices with odd/even least gap.

A366128 gives the least non-subset-sum of prime indices.

Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

Sequence in context: A356680 A356677 A309011 * A086713 A275730 A049771

Adjacent sequences: A365918 A365919 A365920 * A365922 A365923 A365924

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 30 2023

Status

approved