OFFSET
0,3
COMMENTS
The Heinz numbers of these partitions are given by A325131. - Gus Wiseman, Apr 02 2019
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100
EXAMPLE
From Gus Wiseman, Apr 02 2019: (Start)
The a(2) = 2 through a(9) = 7 partitions:
(2) (3) (4) (5) (6) (7) (8) (9)
(11) (111) (1111) (32) (33) (43) (44) (54)
(11111) (42) (52) (53) (63)
(222) (1111111) (62) (72)
(111111) (2222) (432)
(3311) (222111)
(11111111) (111111111)
(End)
MAPLE
b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1, p, select(x-> x<i, m))+
add(`if`(i=j or i in m or j in p, 0, b(n-i*j, i-1,
select(x-> x<=n-i*j, p union {i}),
select(x-> x<i, m union {j}))), j=1..n/i)))
end:
a:= n-> b(n$2, {}$2):
seq(a(n), n=0..40); # Alois P. Heinz, Aug 09 2016
MATHEMATICA
b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #<i&]] + Sum[If[i == j || MemberQ[m, i] || MemberQ[p, j], 0, b[n-i*j, i-1, Select[ p ~Union~ {i}, # <= n-i*j&], Select[m ~Union~ {j}, #<i&]]], {j, 1, n/i}]]]; a[n_] := b[n, n, {}, {}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Length/@Split[#]]=={}&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 18 2006
EXTENSIONS
a(0)=1 prepended and more terms from Alois P. Heinz, Aug 09 2016
STATUS
approved