A064914
Number of ordered biquanimous partitions of 2n.
Original entry on oeis.org
1, 1, 5, 23, 105, 449, 1902, 7828, 31976, 129200, 520425, 2088217, 8371186, 33514797, 134140430, 536699674, 2147154667, 8589198795, 34358341823, 137435830265, 549749857574, 2199010044813, 8796067657649, 35184315676573, 140737380485376, 562949713881526
Offset: 0
From _Gus Wiseman_, Apr 19 2024: (Start)
The a(0) = 1 through a(3) = 23 biquanimous compositions:
() (11) (22) (33)
(112) (123)
(121) (132)
(211) (213)
(1111) (231)
(312)
(321)
(1113)
(1122)
(1131)
(1212)
(1221)
(1311)
(2112)
(2121)
(2211)
(3111)
(11112)
(11121)
(11211)
(12111)
(21111)
(111111)
(End)
The complement is counted by
A371956.
A237258 (aerated) counts biquanimous strict partitions, ranks
A357854.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], MemberQ[Total/@Subsets[#],n]&]],{n,0,5}] (* Gus Wiseman, Apr 19 2024 *)
A371956
Number of non-biquanimous compositions of 2n.
Original entry on oeis.org
0, 1, 3, 9, 23, 63, 146, 364
Offset: 0
The a(1) = 1 through a(3) = 9 compositions:
(2) (4) (6)
(1,3) (1,5)
(3,1) (2,4)
(4,2)
(5,1)
(1,1,4)
(1,4,1)
(2,2,2)
(4,1,1)
The complement is counted by
A064914.
A237258 (aerated) counts biquanimous strict partitions, ranks
A357854.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], !MemberQ[Total/@Subsets[#],n]&]],{n,0,5}]
A372120
Numbers k such that the k-th composition in standard order is biquanimous.
Original entry on oeis.org
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
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.
A237258 (aerated) counts biquanimous strict partitions, ranks
A357854.
-
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]
Showing 1-3 of 3 results.
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