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A365383
Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.
3
1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53
OFFSET
0,2
COMMENTS
Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
22, 15, 19, 17, 20, 17, 20, 16,
22, 17, 20, 17, 21, 17, 20, 17,
22, 17, 20, 17, 21, 17, 20, 17, ...
EXAMPLE
Triangle begins:
1
2 1
3 2 2
5 3 4 3
7 5 6 6 6
11 7 9 8 9 7
15 11 13 13 14 13 14
22 15 19 17 20 17 20 16
30 22 26 26 27 26 28 26 27
42 30 37 34 39 33 40 34 39 34
56 42 50 49 52 50 54 51 54 53 53
77 56 68 64 71 63 73 63 71 65 70 62
101 77 91 89 95 90 97 93 97 97 98 94 99
135 101 122 115 127 115 130 114 131 119 130 117 132 116
176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
(6) (51) (51) (51) (51) (51)
(51) (411) (42) (411) (42) (411)
(42) (321) (411) (33) (411) (321)
(411) (3111) (321) (321) (321) (3111)
(33) (2211) (3111) (3111) (3111) (2211)
(321) (21111) (222) (2211) (222) (21111)
(3111) (111111) (2211) (21111) (2211) (111111)
(222) (21111) (111111) (21111)
(2211) (111111) (111111)
(21111)
(111111)
MATHEMATICA
combu[n_, y_]:=With[{s=Table[{k, i}, {k, Union[y]}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combu[k, #]!={}&]], {n, 0, 12}, {k, 0, n-1}]
CROSSREFS
Column k = 0 is A000041, strict A000009.
The version for subsets is A365381, main diagonal A365376.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Sequence in context: A210795 A210862 A298675 * A144154 A350768 A054710
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 08 2023
STATUS
approved