A098124 -id:A098124 - cp's OEIS frontend

A340657 Numbers with a twice-balanced factorization.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 36, 37, 40, 41, 43, 44, 45, 47, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 88, 89, 92, 97, 98, 99, 100, 101, 103, 104, 107, 109, 113, 116, 117, 120, 124, 127, 131, 135, 136, 137

Offset: 1

Gus Wiseman, Jan 17 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The sequence of terms together with their prime indices begins:
      1: {}            29: {10}          59: {17}
      2: {1}           31: {11}          61: {18}
      3: {2}           36: {1,1,2,2}     63: {2,2,4}
      5: {3}           37: {12}          67: {19}
      7: {4}           40: {1,1,1,3}     68: {1,1,7}
     11: {5}           41: {13}          71: {20}
     12: {1,1,2}       43: {14}          73: {21}
     13: {6}           44: {1,1,5}       75: {2,3,3}
     17: {7}           45: {2,2,3}       76: {1,1,8}
     18: {1,2,2}       47: {15}          79: {22}
     19: {8}           50: {1,3,3}       83: {23}
     20: {1,1,3}       52: {1,1,6}       88: {1,1,1,5}
     23: {9}           53: {16}          89: {24}
     24: {1,1,1,2}     54: {1,2,2,2}     92: {1,1,9}
     28: {1,1,4}       56: {1,1,1,4}     97: {25}
The twice-balanced factorizations of 1920 (with prime indices {1,1,1,1,1,1,1,2,3}) are (8*8*30) and (8*12*20), so 1920 is in the sequence.

The alt-balanced version is A340597.
Positions of nonzero terms in A340655.
The complement is A340656.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]!={}&]

A340598 Number of balanced set partitions of {1..n}.

Original entry on oeis.org

0, 1, 0, 3, 3, 10, 60, 210, 700, 3556, 19845, 105567, 550935, 3120832, 19432413, 127949250, 858963105, 5882733142, 41636699676, 307105857344, 2357523511200, 18694832699907, 152228641035471, 1270386473853510, 10872532998387918, 95531590347525151

Offset: 0

Gus Wiseman, Jan 20 2021

nonn

A set partition is balanced if it has exactly as many blocks as the greatest size of a block.

			The a(1) = 1 through a(5) = 10 balanced set partitions (empty column indicated by dot):
  {{1}}  .  {{1},{2,3}}  {{1,2},{3,4}}  {{1},{2},{3,4,5}}
            {{1,2},{3}}  {{1,3},{2,4}}  {{1},{2,3,4},{5}}
            {{1,3},{2}}  {{1,4},{2,3}}  {{1,2,3},{4},{5}}
                                        {{1},{2,3,5},{4}}
                                        {{1,2,4},{3},{5}}
                                        {{1},{2,4,5},{3}}
                                        {{1,2,5},{3},{4}}
                                        {{1,3,4},{2},{5}}
                                        {{1,3,5},{2},{4}}
                                        {{1,4,5},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unlabeled version is A047993 (A106529).
A000110 counts set partitions.
A000670 counts ordered set partitions.
A113547 counts set partitions by maximin.
Other balance-related sequences:
- A010054 counts balanced strict integer partitions (A002110).
- A098124 counts balanced integer compositions.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A000258, A001055, A006141, A008275, A008277, A008278, A095149.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[#]==Max@@Length/@#&]],{n,0,8}]

\\ D(n,k) counts balanced set partitions with k blocks.
D(n,k)={my(t=sum(i=1, k, x^i/i!) + O(x*x^n)); n!*polcoef(t^k - (t-x^k/k!)^k, n)/k!}
a(n)={sum(k=sqrtint(n), (n+1)\2, D(n,k))} \\ Andrew Howroyd, Mar 14 2021

Terms a(12) and beyond from Andrew Howroyd, Mar 14 2021

A340600 Number of non-isomorphic balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 0, 4, 7, 16, 52, 206, 444, 1624, 5462, 19188, 62890, 215367, 765694, 2854202, 10634247, 39842786, 150669765, 581189458, 2287298588, 9157598354, 37109364812, 151970862472, 629048449881, 2635589433705, 11184718653563, 48064965080106, 208988724514022, 918639253237646, 4079974951494828

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{1,1}}  {{1,1},{1,1}}  {{1},{1},{1,1,1}}
            {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{1,2,2}}
            {{1},{2,3}}  {{1,2},{1,2}}  {{1},{1},{2,2,2}}
            {{2},{1,2}}  {{1,2},{2,2}}  {{1},{1},{2,3,3}}
                         {{1,2},{3,3}}  {{1},{1},{2,3,4}}
                         {{1,2},{3,4}}  {{1},{2},{1,2,2}}
                         {{1,3},{2,3}}  {{1},{2},{2,2,2}}
                                        {{1},{2},{2,3,3}}
                                        {{1},{2},{3,3,3}}
                                        {{1},{2},{3,4,4}}
                                        {{1},{2},{3,4,5}}
                                        {{1},{3},{2,3,3}}
                                        {{1},{4},{2,3,4}}
                                        {{2},{2},{1,2,2}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The version for partitions is A047993.
The co-balanced version is A319616.
The cross-balanced version is A340651.
The twice-balanced version is A340652.
The version for factorizations is A340653.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A098124 counts balanced compositions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
Cf. A001055, A006141, A007717, A316983, A320663, A324522, A340654, A340655.

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

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

A077229 Number of compositions of n where the largest part is less than or equal to the number of parts.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 98, 204, 421, 863, 1766, 3606, 7341, 14913, 30233, 61175, 123589, 249344, 502443, 1011366, 2033894, 4086975, 8206833, 16469875, 33035611, 66234372, 132745859, 265961487, 532717894, 1066778687, 2135822457, 4275459730, 8557335141, 17125445575, 34268965676, 68568213419, 137187103849, 274458924246

Offset: 0

Henry Bottomley, Oct 29 2002

nonn

			a(5)=11 since 5 can be written as 1+1+1+1+1, 1+1+1+2, 1+1+2+1, 1+1+3, 1+2+1+1, 1+2+2, 1+3+1, 2+1+1+1, 2+1+2, 2+2+1, or 3+1+1; but not as 2+3 since then the largest part (3) would be greater than the number of parts (2).

Index entries for sequences related to compositions

Row sums of A077227.

Cf. A064174, A348125.

Table[SeriesCoefficient[1 + Sum[x^k*((1-x^k)/(1-x))^k,{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 01 2014 *)

G.f.: 1 + Sum_{k>=0} ((x^(k+1)-x)/(x-1))^k. - Vladeta Jovovic, Sep 24 2004
G.f.: 1 + Sum_{n>=1} q^n * ( (1-q^n)/(1-q) )^n, the g.f. above, slightly rewritten. [Joerg Arndt, Mar 30 2014]
a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014
a(n) = A098124(n)+A098125(n). - R. J. Mathar, Oct 01 2021

More terms from Vladeta Jovovic, Sep 24 2004

Prepended a(0) = 1, Joerg Arndt, Mar 30 2014

A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350

Offset: 0

Gus Wiseman, Jan 28 2021

nonn

Also the number of integer partitions of n with maximum 2^k where k is the length.

			The partitions for n = 12, 14, 16, 22, 24:
  32211111  32222111  32222221  33333322          33333333
  33111111  33221111  33222211  33333331          4222221111111111
            33311111  33322111  4222111111111111  4322211111111111
                      33331111  4321111111111111  4332111111111111
                                4411111111111111  4422111111111111
                                                  4431111111111111
The conjugate partitions:
  (8,2,2)  (8,3,3)  (8,4,4)  (8,7,7)     (8,8,8)
  (8,3,1)  (8,4,2)  (8,5,3)  (8,8,6)     (16,3,3,2)
           (8,5,1)  (8,6,2)  (16,2,2,2)  (16,4,2,2)
                    (8,7,1)  (16,3,2,1)  (16,4,3,1)
                             (16,4,1,1)  (16,5,2,1)
                                         (16,6,1,1)

Note: A-numbers of Heinz-number sequences are in parentheses below.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
Cf. A003114, A006141, A064174, A098124, A117409, A200750, A340385, A340387, A340599, A340601.

Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]

A348125 Number of compositions of n where the largest part is larger than the number of parts.

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 16, 30, 52, 91, 161, 282, 490, 851, 1471, 2535, 4361, 7483, 12800, 21845, 37210, 63258, 107329, 181775, 307341, 518821, 874492, 1471869, 2473969, 4153018, 6963137, 11661191, 19507566, 32599451, 54423609, 90772692, 151263317, 251849623

Offset: 1

R. J. Mathar, Oct 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A011782, A098124, A098125, A077229.

a(n) + A077229(n) = 2^(n-1).

a(22)-a(38) from Alois P. Heinz, Oct 01 2021

cp's OEIS Frontend

A340657 Numbers with a twice-balanced factorization.

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A340598 Number of balanced set partitions of {1..n}.

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A340600 Number of non-isomorphic balanced multiset partitions of weight n.

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A077229 Number of compositions of n where the largest part is less than or equal to the number of parts.

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A340611 Number of integer partitions of n of length 2^k where k is the greatest part.

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A348125 Number of compositions of n where the largest part is larger than the number of parts.

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