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A372120 Numbers k such that the k-th composition in standard order is biquanimous.

%I A372120 #4 Apr 20 2024 10:51:29
%S A372120 0,3,10,11,13,14,15,36,37,38,39,41,43,44,45,46,47,50,51,52,53,54,55,
%T A372120 57,58,59,60,61,62,63,136,137,138,139,140,141,142,143,145,147,149,150,
%U A372120 151,152,153,154,155,156,157,158,159,162,163,165,166,167,168,169
%N A372120 Numbers k such that the k-th composition in standard order is biquanimous.
%C A372120 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%C A372120 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 A372120 The terms and corresponding compositions begin:
%e A372120    0: ()
%e A372120    3: (1,1)
%e A372120   10: (2,2)
%e A372120   11: (2,1,1)
%e A372120   13: (1,2,1)
%e A372120   14: (1,1,2)
%e A372120   15: (1,1,1,1)
%e A372120   36: (3,3)
%e A372120   37: (3,2,1)
%e A372120   38: (3,1,2)
%e A372120   39: (3,1,1,1)
%e A372120   41: (2,3,1)
%e A372120   43: (2,2,1,1)
%e A372120   44: (2,1,3)
%e A372120   45: (2,1,2,1)
%e A372120   46: (2,1,1,2)
%e A372120   47: (2,1,1,1,1)
%e A372120   50: (1,3,2)
%e A372120   51: (1,3,1,1)
%e A372120   52: (1,2,3)
%e A372120   53: (1,2,2,1)
%e A372120   54: (1,2,1,2)
%t A372120 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A372120 Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]
%Y A372120 These compositions are counted by A064914.
%Y A372120 The unordered version (integer partitions) is A357976, counted by A002219.
%Y A372120 The unordered complement is A371731, counted by A371795, even case A006827.
%Y A372120 The complement is A372119, counted by A371956.
%Y A372120 A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
%Y A372120 A321142 and A371794 count non-biquanimous strict partitions.
%Y A372120 A371791 counts biquanimous sets, differences A232466.
%Y A372120 A371792 counts non-biquanimous sets, differences A371793.
%Y A372120 Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.
%K A372120 nonn
%O A372120 1,2
%A A372120 _Gus Wiseman_, Apr 20 2024