%I #8 May 30 2022 23:35:42
%S 1,0,1,0,1,1,0,2,1,0,0,2,2,1,0,0,3,4,0,0,0,0,4,6,1,0,0,0,0,5,9,1,0,0,
%T 0,0,0,6,11,4,1,0,0,0,0,0,8,20,2,0,0,0,0,0,0,0,10,25,7,0,0,0,0,0,0,0,
%U 0,12,37,6,1,0,0,0,0,0,0,0
%N Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.
%C Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking run-sums (or condensations) until a strict partition is reached. For example, the trajectory of (2,1,1) is (2,1,1) -> (2,2) -> (4).
%C Also the number of integer partitions of n with Kimberling's depth statistic (see A237685, A237750) equal to k-1.
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 2 1 0
%e 0 2 2 1 0
%e 0 3 4 0 0 0
%e 0 4 6 1 0 0 0
%e 0 5 9 1 0 0 0 0
%e 0 6 11 4 1 0 0 0 0
%e 0 8 20 2 0 0 0 0 0 0
%e 0 10 25 7 0 0 0 0 0 0 0
%e 0 12 37 6 1 0 0 0 0 0 0 0
%e 0 15 47 13 2 0 0 0 0 0 0 0 0
%e 0 18 67 15 1 0 0 0 0 0 0 0 0 0
%e 0 22 85 25 3 0 0 0 0 0 0 0 0 0 0
%e 0 27 122 26 1 0 0 0 0 0 0 0 0 0 0 0
%e For example, row n = 8 counts the following partitions (empty columns indicated by dots):
%e . (8) (44) (422) (4211) . . . .
%e (53) (332) (32111)
%e (62) (611) (41111)
%e (71) (2222) (221111)
%e (431) (3221)
%e (521) (3311)
%e (5111)
%e (22211)
%e (311111)
%e (2111111)
%e (11111111)
%t rsn[y_]:=If[y=={},{},NestWhileList[Reverse[Sort[Total/@ Split[Sort[#]]]]&,y,!UnsameQ@@#&]];
%t Table[Length[Select[IntegerPartitions[n],Length[rsn[#]]==k&]],{n,0,15},{k,0,n}]
%Y Row-sums are A000041.
%Y Column k = 1 is A000009.
%Y Column k = 2 is A237685.
%Y Column k = 3 is A237750.
%Y The version for run-lengths instead of run-sums is A225485 or A325280.
%Y This statistic (trajectory length) is ranked by A353841 and A326371.
%Y The version for compositions is A353859, see also A353847-A353858.
%Y A005811 counts runs in binary expansion.
%Y A275870 counts collapsible partitions, ranked by A300273.
%Y A304442 counts partitions with all equal run-sums, ranked by A353833.
%Y A353832 represents the operation of taking run-sums of a partition
%Y A353836 counts partitions by number of distinct run-sums.
%Y A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353840-A353846 pertain to partition run-sum trajectory.
%Y A353845 counts partitions whose run-sum trajectory ends in a singleton.
%Y Cf. A008284, A047966, A181819, A325239, A325277, A353834, A353865.
%K nonn,tabl
%O 0,8
%A _Gus Wiseman_, May 26 2022
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