A382459
Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.
Original entry on oeis.org
1, 1, 0, 2, 1, 3, 2, 7, 4, 10, 19
Offset: 0
The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
The a(1) = 1 through a(7) = 7 multisets:
{1} . {112} {1122} {11123} {111233} {1111234}
{122} {12223} {122233} {1112223}
{12333} {1112333}
{1222234}
{1222333}
{1233334}
{1234444}
Without distinct sums we have
A382458.
Cf.
A000110,
A000670,
A007716,
A255903,
A275780,
A317532,
A321469,
A326519,
A381078,
A381441,
A382428.
-
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]
A382301
Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.
Original entry on oeis.org
1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694
Offset: 0
The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
{4} {5} {6} {7} {8}
{22} {1}{4} {33} {1}{6} {44}
{1}{3} {2}{3} {1}{5} {2}{5} {1}{7}
{11}{3} {2}{4} {3}{4} {2}{6}
{1}{22} {11}{4} {11}{5} {3}{5}
{2}{111} {11}{22} {1}{33} {11}{6}
{1}{2}{3} {3}{22} {2}{33}
{1}{11}{3} {1}{2}{4} {11}{33}
{3}{1111} {11}{222}
{1}{2}{5}
{1}{3}{4}
{1}{3}{22}
{1}{4}{111}
{1}{111}{22}
For distinct blocks instead of block-sums we have
A000726, ranks
A004709.
Twice-partitions of this type (constant with distinct) are counted by
A279786.
The Heinz numbers of these partitions are
A381991, positions of 1 in
A381635.
Normal multiset partitions of this type are counted by
A382203.
For at least one choice we have
A382427.
A050361 counts factorizations into distinct prime powers.
Cf.
A006171,
A047966,
A279784,
A293511,
A295935,
A353864,
A381633,
A381716,
A381990,
A381992,
A381993,
A382079.
-
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]
A382427
Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.
Original entry on oeis.org
1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685
Offset: 0
The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(1111) (221) (51) (61)
(311) (222) (322)
(2111) (321) (331)
(11111) (411) (421)
(2211) (511)
(3111) (2221)
(21111) (4111)
(111111) (22111)
(31111)
(211111)
(1111111)
Twice-partitions of this type (constant with distinct) are counted by
A279786.
Normal multiset partitions of this type are counted by
A382203, sets
A381718.
A050361 counts factorizations into distinct prime powers.
Cf.
A006171,
A047966,
A279784,
A295935,
A300385,
A353864,
A381633,
A381716,
A381990,
A381993,
A382079,
A382876.
-
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]
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