A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050

Offset: 0

Author

Henry Bottomley, Aug 01 2005

Keywords

• nonn

Comments

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

Examples

			a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2.
a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6.
From _Gus Wiseman_, Oct 30 2018: (Start)
The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (2111)   (222)     (331)      (332)
               (1111)  (11111)  (321)     (2221)     (422)
                                (2211)    (3211)     (431)
                                (3111)    (22111)    (2222)
                                (21111)   (31111)    (3221)
                                (111111)  (211111)   (3311)
                                          (1111111)  (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (22)  (32)  (33)   (43)   (44)
            (31)  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (222)  (322)  (71)
                        (321)  (331)  (332)
                        (411)  (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (2222)
                                      (3221)
                                      (3311)
                                      (4211)
                                      (5111)
The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2},{1,2,3,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
(End)
		

Crossrefs

Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921.

Programs

Maple

A000070 := proc(n) add( combinat[numbpart](i),i=0..n) ; end proc:
A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011

Mathematica

f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multhyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multhyp[#]!={}&]],{n,8}] (* Gus Wiseman, Oct 30 2018 *)

PARI

a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018

Formula

a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2).

a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018