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A110618
Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.
27
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
0,5
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
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
EXAMPLE
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)
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 *)
PROG
(PARI) a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018
KEYWORD
nonn
AUTHOR
Henry Bottomley, Aug 01 2005
STATUS
approved