cp's OEIS Frontend

A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

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