A340655 -id:A340655 - cp's OEIS frontend

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

1, 1, 2, 4, 11, 26, 77, 220, 677, 2098, 6756, 22101, 74264, 253684, 883795, 3130432, 11275246, 41240180, 153117873, 576634463, 2201600769, 8517634249, 33378499157, 132438117118, 531873247805, 2161293783123, 8883906870289, 36928576428885, 155196725172548, 659272353608609, 2830200765183775

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,2}}    {{1,2},{1,2}}
                    {{1},{1},{1}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The co-balanced version is A319616.
The balanced version is A340600.
The twice-balanced version is A340652.
The version for factorizations is A340654.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n, G(k,n,k) - G(k-1,n,k) - G(k,n,k-1) + G(k-1,n,k-1)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A340690 Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.

Original entry on oeis.org

2, 8, 12, 16, 32, 48, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 224, 240, 256, 280, 288, 320, 336, 384, 392, 432, 448, 480, 512, 576, 640, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1008, 1024, 1056, 1120, 1152, 1200, 1248, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

			The initial terms and a valid factorization of each:
      2 = 2           168 = 3*7*8        512 = 2*2*2*2*32
      8 = 2*4         192 = 2*2*3*16     576 = 2*2*9*16
     12 = 3*4         200 = 5*5*8        640 = 2*2*10*16
     16 = 4*4         224 = 4*7*8        672 = 2*3*7*16
     32 = 2*2*8       240 = 5*6*8        704 = 2*2*11*16
     48 = 2*3*8       256 = 2*2*4*16     720 = 3*3*5*16
     64 = 2*4*8       280 = 5*7*8        768 = 2*2*2*3*32
     72 = 3*3*8       288 = 2*3*3*16     800 = 2*5*5*16
     80 = 2*5*8       320 = 2*2*5*16     832 = 2*2*13*16
     96 = 2*6*8       336 = 6*7*8        864 = 2*3*9*16
    112 = 2*7*8       384 = 2*2*6*16     896 = 2*2*14*16
    120 = 3*5*8       392 = 7*7*8        960 = 2*2*15*16
    128 = 2*2*2*16    432 = 3*3*3*16    1008 = 3*3*7*16
    144 = 3*6*8       448 = 2*2*7*16    1024 = 2*2*2*4*32
    160 = 4*5*8       480 = 2*3*5*16    1056 = 2*3*11*16

Partitions of the prescribed type are counted by A340611.
The conjugate version is A340689.
A001055 counts factorizations, with strict case A045778.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf. A106529, A117409, A200750, A325134, A340386, A340387, A340599, A340607, A340654, A340655, A340656, A340657.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],2^Length[#]==Max@@#&]!={}&]

cp's OEIS Frontend

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions