login
A367108
Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.
0
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
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.
Sequence in context: A082860 A342859 A342385 * A283845 A365543 A058071
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 18 2023
STATUS
approved