OFFSET
0,4
COMMENTS
An anti-run is a sequence with no adjacent equal parts.
These are partitions with no part appearing more than twice and greatest part appearing only once.
Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024
EXAMPLE
The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(311) (321) (61) (71) (72)
(411) (322) (422) (81)
(421) (431) (432)
(511) (521) (522)
(3211) (611) (531)
(3221) (621)
(4211) (711)
(4221)
(4311)
(5211)
(32211)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#, UnsameQ]&]], {n, 0, 30}]
PROG
(PARI)
A_x(N) = {my(x='x+O('x^N), f=sum(i=0, N, (x^i)*prod(j=1, i-1, (1-x^(3*j))/(1-x^j)))); Vec(f)}
A_x(51) \\ John Tyler Rascoe, Aug 21 2024
CROSSREFS
Includes all strict partitions A000009.
For compositions instead of partitions we have A374761.
A000041 counts integer partitions.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 14 2024
STATUS
approved