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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

%I #8 Jun 04 2022 22:27:26

%S 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,

%T 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,

%U 2,13,98,191,79,4,2,114,261,105,8,14,148,282,164,19

%N 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.

%C 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).

%e Triangle begins:

%e 1

%e 1

%e 2

%e 2 1

%e 4 1

%e 2 5

%e 5 5 1

%e 2 12 1

%e 8 11 3

%e 3 19 8

%e 5 27 9 1

%e 2 34 19 1

%e 15 26 34 2

%e 2 49 45 5

%e 5 68 48 14

%e 4 58 98 15 1

%e For example, row n = 8 counts the following partitions:

%e (8) (53) (431)

%e (44) (62) (521)

%e (422) (71) (3221)

%e (2222) (332)

%e (4211) (611)

%e (41111) (3311)

%e (221111) (5111)

%e (11111111) (22211)

%e (32111)

%e (311111)

%e (2111111)

%t Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]

%Y Row sums are A000041.

%Y Row-lengths are A003056.

%Y The last part of the same trajectory is A353842.

%Y Column k = 1 is A353845, compositions A353858.

%Y The length of the trajectory is A353846.

%Y The version for compositions is A353856.

%Y A275870 counts collapsible partitions, ranked by A300273.

%Y A304442 counts partitions with constant run-sums, ranked by A353833/A353834.

%Y A325268 counts partitions by omicron, rank statistic A304465.

%Y A353837 counts partitions with all distinct run-sums, ranked by A353838.

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y A353847 represents the run-sums of a composition, partitions A353832.

%Y A353864 counts rucksack partitions, ranked by A353866.

%Y A353865 counts perfect rucksack partitions, ranked by A353867.

%Y A353932 lists run-sums of standard compositions.

%Y Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

%Y Cf. A047966, A237685, A325277, A353841, A353853-A353859.

%K nonn,tabf

%O 0,3

%A _Gus Wiseman_, Jun 04 2022