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%I #17 Jul 07 2020 06:05:58
%S 1,0,2,2,2,3,5,4,7,7,13,16,19,23,33,34,44,58,63,80,101,112,139,171,
%T 196,234,288,328,394,478,545,658,777,881,1050,1236,1414,1666,1936,
%U 2216,2592,3018,3428,3992,4604,5243,6069,6986,7951,9139,10447,11892,13625
%N Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.
%C The Heinz numbers of these partitions are given by A325131. - _Gus Wiseman_, Apr 02 2019
%H Alois P. Heinz, <a href="/A114639/b114639.txt">Table of n, a(n) for n = 0..100</a>
%e From _Gus Wiseman_, Apr 02 2019: (Start)
%e The a(2) = 2 through a(9) = 7 partitions:
%e (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (111) (1111) (32) (33) (43) (44) (54)
%e (11111) (42) (52) (53) (63)
%e (222) (1111111) (62) (72)
%e (111111) (2222) (432)
%e (3311) (222111)
%e (11111111) (111111111)
%e (End)
%p b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1, p, select(x-> x<i, m))+
%p add(`if`(i=j or i in m or j in p, 0, b(n-i*j, i-1,
%p select(x-> x<=n-i*j, p union {i}),
%p select(x-> x<i, m union {j}))), j=1..n/i)))
%p end:
%p a:= n-> b(n$2, {}$2):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 09 2016
%t 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_ *)
%t Table[Length[Select[IntegerPartitions[n],Intersection[#,Length/@Split[#]]=={}&]],{n,0,30}] (* _Gus Wiseman_, Apr 02 2019 *)
%Y Cf. A052335, A087153, A114640, A115584, A117144, A276429, A324572, A325130, A325131, A336032.
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Feb 18 2006
%E a(0)=1 prepended and more terms from _Alois P. Heinz_, Aug 09 2016
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