OFFSET
0,3
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). The first partition whose run-sums are not all distinct is (2,1,1).
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..100
EXAMPLE
The a(0) = 1 through a(6) = 10 partitions:
() (1) (2) (3) (4) (5) (6)
(11) (21) (22) (32) (33)
(111) (31) (41) (42)
(1111) (221) (51)
(311) (222)
(2111) (321)
(11111) (411)
(2211)
(21111)
(111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]], {n, 0, 15}]
PROG
(Sage) a353837 = lambda n: sum( abs(BipartiteGraph( Matrix(len(p), len(D:=list(set.union(*map(lambda t: set(divisors(t)), p)))), lambda i, j: p[i]%D[j]==0) ).matching_polynomial()[len(D)-len(p)]) for p in Partitions(n, max_slope=-1) ) # Max Alekseyev, Sep 11 2023
CROSSREFS
A005811 counts runs in binary expansion.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353849 counts distinct run-sums in standard compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 26 2022
STATUS
approved