OFFSET
0,7
LINKS
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.
Central diagonal n = 2k is A126796 also.
This is the triangle for the rank statistic A365920.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A073491 lists numbers with gap-free prime indices.
A366128 gives the least non-subset-sum of prime indices.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 30 2023
STATUS
approved