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A353843
Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.
3
1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19
OFFSET
0,3
COMMENTS
The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).
EXAMPLE
Triangle begins:
1
1
2
2 1
4 1
2 5
5 5 1
2 12 1
8 11 3
3 19 8
5 27 9 1
2 34 19 1
15 26 34 2
2 49 45 5
5 68 48 14
4 58 98 15 1
For example, row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(422) (71) (3221)
(2222) (332)
(4211) (611)
(41111) (3311)
(221111) (5111)
(11111111) (22211)
(32111)
(311111)
(2111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&, #]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Row sums are A000041.
Row-lengths are A003056.
The last part of the same trajectory is A353842.
Column k = 1 is A353845, compositions A353858.
The length of the trajectory is A353846.
The version for compositions is A353856.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
A325268 counts partitions by omicron, rank statistic A304465.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
A353932 lists run-sums of standard compositions.
Sequence in context: A197250 A275578 A112085 * A090002 A061298 A276468
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jun 04 2022
STATUS
approved