A114638 - OEIS

A114638

Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

%I #28 Oct 27 2019 20:58:25

%S 1,1,0,0,2,1,1,0,2,2,3,5,5,6,9,7,8,14,12,16,21,28,32,43,47,61,68,84,

%T 89,109,126,140,170,198,227,261,323,362,427,501,581,658,794,880,1036,

%U 1175,1355,1526,1776,1985,2281,2588,2943,3312,3799,4271,4852,5497

%N Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

%C The Heinz numbers of these integer partitions are given by A324570. - _Gus Wiseman_, Mar 09 2019

%H Alois P. Heinz, <a href="/A114638/b114638.txt">Table of n, a(n) for n = 0..500</a>

%e a(10) = 3 because we have [5,1,1,1,1,1], [3,3,3,1] and [3,2,2,1,1,1].

%e From _Gus Wiseman_, Mar 09 2019: (Start)

%e The a(1) = 1 through a(12) = 5 integer partitions (empty columns not shown):

%e 1 22 221 3111 3311 333 3331 32222 33222

%e 211 41111 321111 322111 44111 322221

%e 511111 322211 332211

%e 332111 4221111

%e 4211111 6111111

%e (End)

%p a:=proc(n) local P,c,j,S: with(combinat): P:=partition(n): c:=0: for j from 1 to nops(P) do S:=convert(P[j],set): if nops(P[j])=sum(S[i],i=1..nops(S)) then c:=c+1 else c:=c fi: c: od: end: seq(a(n), n=0..35); # _Emeric Deutsch_, Mar 01 2006

%t a[n_] := Module[{P, c, j, S}, P = IntegerPartitions[n]; c = 0; For[j = 1, j <= Length[P], j++, S = Union[P[[j]]]; If[Length[P[[j]]] == Total[S], c++] ]; c];

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 07 2018, after _Emeric Deutsch_ *)

%o (PARI) apply( A114638(n,s=0)={forpart(p=n,#p==vecsum(Set(p))&&s++); s}, [0..50]) \\ _M. F. Hasler_, Oct 27 2019

%Y Cf. A003114, A006141, A039900, A047993, A064174, A066328, A243149 (the same for compositions).

%Y Cf. A324516, A324518, A324520, A324521, A324570.

%Y Cf. A116861 (number of partitions of n having a given sum of distinct parts).

%K nonn

%O 0,5

%A _Vladeta Jovovic_, Feb 18 2006

%E More terms from _Emeric Deutsch_, Mar 01 2006