|
%I #9 May 09 2021 16:32:53
%S 1,1,2,3,4,4,6,6,9,10,12,12,17,16,20,25,27,29,35,39,44,57,53,66,75,84,
%T 84,114,112,131,133,162,167,209,192,242,250,289,279,363,348,417,404,
%U 502,487,608,557,706,682,835,773,1004,922,1149,1059,1344,1257,1595
%N Number of uniform knapsack partitions of n.
%C An integer partition is uniform if all parts appear with the same multiplicity, and knapsack if every distinct submultiset has a different sum.
%H Fausto A. C. Cariboni, <a href="/A326035/b326035.txt">Table of n, a(n) for n = 0..650</a>
%e The a(1) = 1 through a(8) = 9 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (11111) (51) (61) (62)
%e (222) (421) (71)
%e (111111) (1111111) (521)
%e (2222)
%e (3311)
%e (11111111)
%t sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
%t ks[n_]:=Select[IntegerPartitions[n],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
%t Table[Length[Select[ks[n],SameQ@@Length/@Split[#]&]],{n,30}]
%Y Cf. A002033, A047966, A072774, A108917, A275972, A276024, A299702.
%Y Cf. A325592, A325858, A326015, A326016, A326017, A326036, A326037.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jun 04 2019
|
