A371794 -id:A371794 - cp's OEIS frontend

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

0, 3, 10, 11, 13, 14, 15, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 136, 137, 138, 139, 140, 141, 142, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 165, 166, 167, 168, 169

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

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.

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.

			The terms and corresponding compositions begin:
   0: ()
   3: (1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  36: (3,3)
  37: (3,2,1)
  38: (3,1,2)
  39: (3,1,1,1)
  41: (2,3,1)
  43: (2,2,1,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  50: (1,3,2)
  51: (1,3,1,1)
  52: (1,2,3)
  53: (1,2,2,1)
  54: (1,2,1,2)

These compositions are counted by A064914.
The unordered version (integer partitions) is A357976, counted by A002219.
The unordered complement is A371731, counted by A371795, even case A006827.
The complement is A372119, counted by A371956.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152

Offset: 1

Gus Wiseman, Apr 13 2024

			The terms together with their binary expansions and binary indices begin:
        1:                1 ~ {1}
        2:               10 ~ {2}
        4:              100 ~ {3}
        8:             1000 ~ {4}
       12:             1100 ~ {3,4}
       16:            10000 ~ {5}
       32:           100000 ~ {6}
       64:          1000000 ~ {7}
      128:         10000000 ~ {8}
      144:         10010000 ~ {5,8}
      256:        100000000 ~ {9}
      288:        100100000 ~ {6,9}
      512:       1000000000 ~ {10}
      576:       1001000000 ~ {7,10}
     1024:      10000000000 ~ {11}
     2048:     100000000000 ~ {12}
     3072:     110000000000 ~ {11,12}
     4096:    1000000000000 ~ {13}
     8192:   10000000000000 ~ {14}
    16384:  100000000000000 ~ {15}
    32768: 1000000000000000 ~ {16}
    32800: 1000000000100000 ~ {6,16}

For prime instead of binary indices we have A326534.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A321142 and A371794 count non-biquanimous strict partitions.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A357976 ranks the biquanimous partitions counted by A002219 aerated.
A371731 ranks the non-biquanimous partitions counted by A371795, A006827.
Cf. A035470, A038041, A237258, A320324, A321453, A321455, A326518, A336137, A371783, A371791, A371796.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]

cp's OEIS Frontend

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

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