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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.

%I A353843 #8 Jun 04 2022 22:27:26
%S A353843 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 A353843 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 A353843 2,13,98,191,79,4,2,114,261,105,8,14,148,282,164,19
%N 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.
%C A353843 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 A353843 Triangle begins:
%e A353843    1
%e A353843    1
%e A353843    2
%e A353843    2  1
%e A353843    4  1
%e A353843    2  5
%e A353843    5  5  1
%e A353843    2 12  1
%e A353843    8 11  3
%e A353843    3 19  8
%e A353843    5 27  9  1
%e A353843    2 34 19  1
%e A353843   15 26 34  2
%e A353843    2 49 45  5
%e A353843    5 68 48 14
%e A353843    4 58 98 15  1
%e A353843 For example, row n = 8 counts the following partitions:
%e A353843   (8)         (53)       (431)
%e A353843   (44)        (62)       (521)
%e A353843   (422)       (71)       (3221)
%e A353843   (2222)      (332)
%e A353843   (4211)      (611)
%e A353843   (41111)     (3311)
%e A353843   (221111)    (5111)
%e A353843   (11111111)  (22211)
%e A353843               (32111)
%e A353843               (311111)
%e A353843               (2111111)
%t A353843 Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]
%Y A353843 Row sums are A000041.
%Y A353843 Row-lengths are A003056.
%Y A353843 The last part of the same trajectory is A353842.
%Y A353843 Column k = 1 is A353845, compositions A353858.
%Y A353843 The length of the trajectory is A353846.
%Y A353843 The version for compositions is A353856.
%Y A353843 A275870 counts collapsible partitions, ranked by A300273.
%Y A353843 A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
%Y A353843 A325268 counts partitions by omicron, rank statistic A304465.
%Y A353843 A353837 counts partitions with all distinct run-sums, ranked by A353838.
%Y A353843 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353843 A353847 represents the run-sums of a composition, partitions A353832.
%Y A353843 A353864 counts rucksack partitions, ranked by A353866.
%Y A353843 A353865 counts perfect rucksack partitions, ranked by A353867.
%Y A353843 A353932 lists run-sums of standard compositions.
%Y A353843 Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
%Y A353843 Cf. A047966, A237685, A325277, A353841, A353853-A353859.
%K A353843 nonn,tabf
%O A353843 0,3
%A A353843 _Gus Wiseman_, Jun 04 2022