A371956 - OEIS

%I #5 Apr 20 2024 10:51:38

%S 0,1,3,9,23,63,146,364

%N Number of non-biquanimous compositions of 2n.

%C A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

%e The a(1) = 1 through a(3) = 9 compositions:

%e   (2)  (4)    (6)

%e        (1,3)  (1,5)

%e        (3,1)  (2,4)

%e               (4,2)

%e               (5,1)

%e               (1,1,4)

%e               (1,4,1)

%e               (2,2,2)

%e               (4,1,1)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], !MemberQ[Total/@Subsets[#],n]&]],{n,0,5}]

%Y The unordered complement is A002219, ranks A357976.

%Y The unordered version is A006827, even case of A371795, ranks A371731.

%Y The complement is counted by A064914.

%Y These compositions have ranks A372119, complement A372120.

%Y A237258 (aerated) counts biquanimous strict partitions, ranks A357854.

%Y A321142 and A371794 count non-biquanimous strict partitions.

%Y A371791 counts biquanimous sets, differences A232466.

%Y A371792 counts non-biquanimous sets, differences A371793.

%Y Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Apr 20 2024