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Numbers k such that the k-th composition in standard order is biquanimous.
3

%I #4 Apr 20 2024 10:51:29

%S 0,3,10,11,13,14,15,36,37,38,39,41,43,44,45,46,47,50,51,52,53,54,55,

%T 57,58,59,60,61,62,63,136,137,138,139,140,141,142,143,145,147,149,150,

%U 151,152,153,154,155,156,157,158,159,162,163,165,166,167,168,169

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

%C 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 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 terms and corresponding compositions begin:

%e 0: ()

%e 3: (1,1)

%e 10: (2,2)

%e 11: (2,1,1)

%e 13: (1,2,1)

%e 14: (1,1,2)

%e 15: (1,1,1,1)

%e 36: (3,3)

%e 37: (3,2,1)

%e 38: (3,1,2)

%e 39: (3,1,1,1)

%e 41: (2,3,1)

%e 43: (2,2,1,1)

%e 44: (2,1,3)

%e 45: (2,1,2,1)

%e 46: (2,1,1,2)

%e 47: (2,1,1,1,1)

%e 50: (1,3,2)

%e 51: (1,3,1,1)

%e 52: (1,2,3)

%e 53: (1,2,2,1)

%e 54: (1,2,1,2)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

%Y These compositions are counted by A064914.

%Y The unordered version (integer partitions) is A357976, counted by A002219.

%Y The unordered complement is A371731, counted by A371795, even case A006827.

%Y The complement is A372119, counted by A371956.

%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

%O 1,2

%A _Gus Wiseman_, Apr 20 2024