%I #7 Sep 25 2023 12:56:02

%S 1,1,0,1,1,0,2,0,1,0,2,1,1,1,0,4,0,2,0,1,0,5,1,0,3,1,1,0,8,0,3,0,3,0,

%T 1,0,10,2,1,2,2,3,1,1,0,16,0,5,0,3,0,5,0,1,0,20,2,2,4,2,6,0,4,1,1,0,

%U 31,0,6,0,8,0,5,0,5,0,1,0,39,4,4,4,1,6,6,3,2,6,1,1,0

%N Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

%C For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

%e The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 2 0 1 0

%e 2 1 1 1 0

%e 4 0 2 0 1 0

%e 5 1 0 3 1 1 0

%e 8 0 3 0 3 0 1 0

%e 10 2 1 2 2 3 1 1 0

%e 16 0 5 0 3 0 5 0 1 0

%e 20 2 2 4 2 6 0 4 1 1 0

%e 31 0 6 0 8 0 5 0 5 0 1 0

%e 39 4 4 4 1 6 6 3 2 6 1 1 0

%e 55 0 13 0 8 0 12 0 6 0 6 0 1 0

%e 71 5 8 7 3 5 3 16 3 6 0 6 1 1 0

%e Row n = 6 counts the following partitions:

%e (321) (411) . (51) (33) (6) .

%e (3111) (42)

%e (2211) (222)

%e (21111)

%e (111111)

%t Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]

%Y Row sums are A000041.

%Y The rank statistic counted by this triangle is A325799.

%Y The strict case is A365545, weighted row sums A365922.

%Y The complement (positive subset-sum) is A365658.

%Y Weighted row sums are A365918, for positive subset-sums A304792.

%Y A046663 counts partitions w/o a submultiset summing to k, strict A365663.

%Y A126796 counts complete partitions, ranks A325781, strict A188431.

%Y A364350 counts combination-free strict partitions, complement A364839.

%Y A365543 counts partitions with a submultiset summing to k, strict A365661.

%Y A365924 counts incomplete partitions, ranks A365830, strict A365831.

%Y Cf. A000009, A006827, A122768, A364272, A365919, A365921.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Sep 24 2023