A004250 -id:A004250 - cp's OEIS frontend

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) partitions for selected n (A = 10):
  n=9:   n=12:   n=21:      n=24:       n=30:
------------------------------------------------------
  (333)  (444)   (777)      (888)       (AAA)
         (3333)  (444333)   (6666)      (66666)
                 (3333333)  (444444)    (555555)
                            (555333)    (666444)
                            (4443333)   (777333)
                            (33333333)  (6663333)
                                        (55533333)
                                        (444333333)
                                        (3333333333)

The version for only parts > 2 is A008483.
The version for only multiplicities > 2 is A100405.
The version for parts and multiplicities > 1 is A339222, ranked by A062739.
For prime parts and multiplicities we have A351982, compositions A353429.
The version for compositions is A353428 (partial A078012, A353400).
These partitions are ranked by A353502.
A000726 counts partitions with all mults <= 2, compositions A128695.
A004250 counts partitions with some part > 2, compositions A008466.
A137200 counts compositions with all parts and run-lengths <= 2.
Cf. A003242, A047966, A055923, A098859, A114901, A325534, A333755, A335464.

Table[Length[Select[IntegerPartitions[n],Min@@#>2&&Min@@Length/@Split[#]>2&]],{n,0,30}]

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Gus Wiseman, May 23 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

A029893 Number of graphical partitions with up to n parts (?).

Original entry on oeis.org

1, 2, 4, 10, 24, 68, 198, 656, 2112

Offset: 1

torsten.sillke(AT)lhsystems.com

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

A possible duplicate of A028506.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

Original entry on oeis.org

27, 45, 54, 63, 75, 81, 90, 99, 105, 108, 117, 126, 135, 147, 150, 153, 162, 165, 171, 180, 189, 195, 198, 207, 210, 216, 225, 231, 234, 243, 252, 255, 261, 270, 273, 279, 285, 294, 297, 300, 306, 315, 324, 330, 333, 342, 345, 351, 357, 360, 363, 369, 378, 387

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A006918 (see Emeric Deutsch's comment there).

			The sequence of terms together with their prime indices begins:
   27: {2,2,2}
   45: {2,2,3}
   54: {1,2,2,2}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   90: {1,2,2,3}
   99: {2,2,5}
  105: {2,3,4}
  108: {1,1,2,2,2}
  117: {2,2,6}
  126: {1,2,2,4}
  135: {2,2,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  153: {2,2,7}
  162: {1,2,2,2,2}
  165: {2,3,5}
  171: {2,2,8}
  180: {1,1,2,2,3}

Cf. A000726, A002620, A004250, A006918, A056239, A097701, A112798, A257990, A297113, A325164, A325169, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]>=3&&Reverse[primeMS[#]][[3]]==2&]

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A080257.

Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

Cf. A000041, A002133, A004250.

A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022

A334827 The number of oriented star-like and star trees with n arcs.

Original entry on oeis.org

4, 17, 66, 221, 688, 2034, 5788, 15998, 43192, 114496, 298712, 769340, 1959064, 4940761, 12354210, 30660947, 75583868, 185208833, 451356846, 1094522547, 2642121008, 6351335083, 15208854510, 36288478177, 86295204732, 204571273167, 483532711338, 1139738858221

Offset: 3

R. J. Mathar, Jun 09 2020

nonn

			a(6)=221 counts 132 oriented star-like trees with 3 rays and 6 arcs, 62 with 4 rays and 6 arcs, 20 with 5 rays and 6 arcs, and 7 star trees. In the illustrations in A000238 [Mathar] this is the same as 48 (shape 2) + 64 (shape 3) + 20 (shape 4) +32 (shape 7) + 30 (shape 8) +20 (shape 10) + 7 (shape 11).

Wikipedia, Starlike Tree
Index entries for sequences related to trees

Cf. A000238 (oriented trees), A051437 (linear oriented trees), A209406 (star-like oriented by number of arcs and rays), A004250 (undirected edges).

a(n) = A034899(n) -2^(n+1) = Sum_{k>=3} A209406(n,k).

A029892 Number of even graphical partitions of order 2n - number of odd graphical partitions of order 2n.

Original entry on oeis.org

1, 3, 8, 27, 88, 313, 1095, 4007, 14511

Offset: 1

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

The graphical partitions considered here are for graphs with 2n vertices and with half-loops allowed. Half-loops are loops which count as 1 towards the degree of the vertex. See A029889 for additional information. - Andrew Howroyd, Jan 11 2024

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889, A029890, A029891.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

a(n) = A029891(2*n) - A029890(2*n). - Andrew Howroyd, Jan 10 2024

A084842 Number of rooted trees with n nodes with a height of 2 and with at least 1 node at height 1 has degree > 2.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 37

Offset: 4

Jon Perry, Jul 12 2003

nonn

For n=9 we have the following valid graphic partitions; 9,82,73,64,55,443,533,542,632,722,3333,4332,4422,5322,43222. The basic pattern is partitions of n+k into k+1 parts, minimum part 2. After checking a graph can be produced (e.g. 6222 cannot), adding the number of distinct elements in each pattern gives the sequence, except for (n-1)2, which is always 1 and only counting elements which are greater than or equal to the number of elements in a pattern (e.g. 722 only yields 1 possibility). So the patterns above yield 1,1,2,2,1,2,2,3,3,1,1,2,1,2,1, adding gives a(9)=25

Cf. A004250.

cp's OEIS Frontend

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A029893 Number of graphical partitions with up to n parts (?).

Views

Author

Keywords

References

Links

Crossrefs

Formula

A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A334827 The number of oriented star-like and star trees with n arcs.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A029892 Number of even graphical partitions of order 2n - number of odd graphical partitions of order 2n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A084842 Number of rooted trees with n nodes with a height of 2 and with at least 1 node at height 1 has degree > 2.

Views

Author

Keywords

Comments

Crossrefs