A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

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, 89, 109, 126, 140, 170, 198, 227, 261, 323, 362, 427, 501, 581, 658, 794, 880, 1036, 1175, 1355, 1526, 1776, 1985, 2281, 2588, 2943, 3312, 3799, 4271, 4852, 5497

Offset: 0

Author

Vladeta Jovovic, Feb 18 2006

Keywords

• nonn

Comments

The Heinz numbers of these integer partitions are given by A324570. - Gus Wiseman, Mar 09 2019

Examples

			a(10) = 3 because we have [5,1,1,1,1,1], [3,3,3,1] and [3,2,2,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(12) = 5 integer partitions (empty columns not shown):
  1  22   221  3111  3311   333     3331    32222    33222
     211             41111  321111  322111  44111    322221
                                    511111  322211   332211
                                            332111   4221111
                                            4211111  6111111
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

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

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

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

Programs

Maple

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

Mathematica

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];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 07 2018, after Emeric Deutsch *)

PARI

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

Extensions

More terms from Emeric Deutsch, Mar 01 2006