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

0,5

Comments

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

Links

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

Example

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)

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 *)

Prog

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

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).

Sequence in context: A331410 A336928 A366388 * A123340 A360455 A267486

Adjacent sequences: A114635 A114636 A114637 * A114639 A114640 A114641

Keyword

nonn

Author

Vladeta Jovovic, Feb 18 2006

Extensions

More terms from Emeric Deutsch, Mar 01 2006

Status

approved