OFFSET
0,5
COMMENTS
The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).
EXAMPLE
Triangle begins:
1
0 1
0 2 0
0 2 1 0
0 4 1 0 0
0 2 5 0 0 0
0 5 5 1 0 0 0
0 2 12 1 0 0 0 0
0 7 12 3 0 0 0 0 0
0 3 19 8 0 0 0 0 0 0
0 5 27 9 1 0 0 0 0 0 0
0 2 33 20 1 0 0 0 0 0 0 0
0 13 28 34 2 0 0 0 0 0 0 0 0
0 2 48 46 5 0 0 0 0 0 0 0 0 0
0 5 65 51 14 0 0 0 0 0 0 0 0 0 0
0 4 57 99 15 1 0 0 0 0 0 0 0 0 0 0
For example, row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(422) (71) (3221)
(2222) (332)
(41111) (611)
(221111) (3311)
(11111111) (4211)
(5111)
(22211)
(32111)
(311111)
(2111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Split[#]]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 26 2022
STATUS
approved