OFFSET
0,3
COMMENTS
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - Andrew Howroyd, May 04 2023
EXAMPLE
The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (221) (33) (322)
(211) (311) (222) (331)
(1111) (2111) (411) (511)
(11111) (3111) (2221)
(21111) (3211)
(111111) (4111)
(22111)
(31111)
(211111)
(1111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Commonest[#]]==1&]], {n, 0, 30}]
PROG
(PARI) seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 30 2023
STATUS
approved