A000569 -id:A000569 - cp's OEIS frontend

A320924 Heinz numbers of multigraphical partitions.

1, 4, 9, 12, 16, 25, 27, 30, 36, 40, 48, 49, 63, 64, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 144, 147, 154, 160, 165, 169, 175, 189, 192, 196, 198, 210, 220, 225, 243, 250, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.
Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

			The sequence of all multigraphical partitions begins: (), (11), (22), (211), (1111), (33), (222), (321), (2211), (3111), (21111), (44), (422), (111111), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (221111).
From _Gus Wiseman_, May 23 2021: (Start)
The sequence of terms together with their prime indices and a multigraph realizing each begins:
    1:      () | {}
    4:    (11) | {{1,2}}
    9:    (22) | {{1,2},{1,2}}
   12:   (112) | {{1,3},{2,3}}
   16:  (1111) | {{1,2},{3,4}}
   25:    (33) | {{1,2},{1,2},{1,2}}
   27:   (222) | {{1,2},{1,3},{2,3}}
   30:   (123) | {{1,3},{2,3},{2,3}}
   36:  (1122) | {{1,2},{3,4},{3,4}}
   40:  (1113) | {{1,4},{2,4},{3,4}}
   48: (11112) | {{1,2},{3,5},{4,5}}
   49:    (44) | {{1,2},{1,2},{1,2},{1,2}}
   63:   (224) | {{1,3},{1,3},{2,3},{2,3}}
(End)

These partitions are counted by A209816.
The case with odd weights is A322109.
The conjugate case of equality is A340387.
The conjugate version with odd weights allowed is A344291.
The conjugate opposite version is A344292.
The opposite version with odd weights allowed is A344296.
The conjugate version is A344413.
The conjugate opposite version with odd weights allowed is A344414.
The case of equality is A344415.
The opposite version is A344416.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.
A334201 adds up all prime indices except the greatest.
Cf. A000041, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]!={}&]

Members m of A300061 such that A061395(m) <= A056239(m)/2. - Gus Wiseman, May 23 2021

A004250 Number of partitions of n into 3 or more parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, 127, 168, 222, 288, 375, 480, 616, 781, 990, 1243, 1562, 1945, 2422, 2996, 3703, 4550, 5588, 6826, 8332, 10126, 12292, 14865, 17958, 21618, 25995, 31165, 37317, 44562

Offset: 1

N. J. A. Sloane

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014

Also the number of graphical partitions of 2n into n parts. - Gus Wiseman, Jan 08 2021

			a(6)=7 because there are three partitions of n=6 with i=3 parts: [4, 1, 1], [3, 2, 1], [2, 2, 2] and two partitions with i=4 parts: [3, 1, 1, 1], [2, 2, 1, 1] and one partition with i=5 parts: [2, 1, 1, 1, 1] and one partition with i=6 parts: [1, 1, 1, 1, 1, 1].
From _Gus Wiseman_, Jan 18 2021: (Start)
The a(3) = 1 through a(7) = 11 graphical partitions of 2n into n parts:
  (222)  (2222)  (22222)  (222222)  (2222222)
         (3221)  (32221)  (322221)  (3222221)
                 (33211)  (332211)  (3322211)
                 (42211)  (333111)  (3332111)
                          (422211)  (4222211)
                          (432111)  (4322111)
                          (522111)  (4331111)
                                    (4421111)
                                    (5222111)
                                    (5321111)
                                    (6221111)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
N. Metropolis and P. R. Stein, The enumeration of graphical partitions, Europ. J. Combin., 1 (1980), 139-1532.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics. Spider Graph
Wikipedia, Starlike tree
Index entries for sequences related to graphical partitions

Cf. A004251, A029889, A035300, A095268, A058984.
Rightmost column of A259873.
Central column of A339659.
A000041 counts partitions of 2n into n parts, ranked by A340387.
A000569 counts graphical partitions, ranked by A320922.
A008284 counts partitions by sum and length.
A027187 counts partitions of even length.
A309356 ranks simple covering graphs.
The following count vertex-degree partitions and give their Heinz numbers:
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339617 counts non-graphical partitions of 2n (A339618).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A000219, A339560, A339561, A339661.
Partial sums of A117995.

with(combinat);
for i from 1 to 15 do pik(i,3) od;
pik:= proc(n::integer, k::integer)
# Thomas Wieder, Jan 30 2007
local i, Liste, Result;
if k > n or n < 0 or k < 1 then
return fail
end if;
Result := 0;
for i from k to n do
Liste:= PartitionList(n,i);
#print(Liste);
Result := Result + nops(Liste);
end do;
return Result;
end proc;
PartitionList := proc (n, k)
# Authors: Herbert S. Wilf and Joanna Nordlicht. Source: Lecture Notes
# "East Side West Side,..." University of Pennsylvania, USA, 2002.
# Available at: http://www.cis.upenn.edu/~wilf/lecnotes.html
# Calculates the partition of n into k parts.
# E.g. PartitionList(5,2) --> [[4, 1], [3, 2]].
local East, West;
if n < 1 or k < 1 or n < k then
RETURN([])
elif n = 1 then
RETURN([[1]])
else if n < 2 or k < 2 or n < k then
West := []
else
West := map(proc (x) options operator, arrow;
[op(x), 1] end proc,PartitionList(n-1,k-1)) end if;
if k <= n-k then
East := map(proc (y) options operator, arrow;
map(proc (x) options operator, arrow; x+1 end proc,y) end proc,PartitionList(n-k,k))
else East := [] end if;
RETURN([op(West), op(East)])
end if;
end proc;
#  Thomas Wieder, Feb 01 2007
ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); # Zerinvary Lajos, Mar 25 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card >=3)},unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # Zerinvary Lajos, Mar 21 2009

Length /@ Table[Select[Partitions[n], Length[#] > 2 &], {n, 20}] (* Eric W. Weisstein, May 16 2007 *)
Table[Count[Length /@ Partitions[n], ?(# > 2 &)], {n, 20}] (* _Eric W. Weisstein, May 16 2017 *)
Table[PartitionsP[n] - Floor[n/2] - 1, {n, 20}] (* Eric W. Weisstein, May 16 2017 *)
Length /@ Table[IntegerPartitions[n, {3, n}], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */

G.f.: Sum_{n>=0} (q^n / Product_{k=1..n+3} (1 - q^k)). -  N. J. A. Sloane
a(n) = A000041(n) - floor((n+2)/2) = A000041(n)-A004526(n+2) = A058984(n)-1. - Vladeta Jovovic, Jun 18 2003
Let P(n,i) denote the number of partitions of n into i parts. Then a(n) = Sum_{i=3..n} P(n,i). - Thomas Wieder, Feb 01 2007
a(n) = A259873(n,n). - Gus Wiseman, Jan 08 2021

Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007

A320922 Heinz numbers of graphical partitions.

Original entry on oeis.org

1, 4, 12, 16, 27, 36, 40, 48, 64, 81, 90, 108, 112, 120, 144, 160, 192, 225, 243, 252, 256, 270, 300, 324, 336, 352, 360, 400, 432, 448, 480, 567, 576, 625, 630, 640, 675, 729, 750, 756, 768, 792, 810, 832, 840, 900, 972, 1000, 1008, 1024, 1056, 1080, 1120

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

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

An integer partition is graphical if it comprises the vertex-degrees of some simple graph.

			The sequence of all graphical partitions begins: (), (11), (211), (1111), (222), (2211), (3111), (21111), (111111), (2222), (3221), (22211), (41111), (32111), (221111), (311111), (2111111), (3322), (22222), (42211).

Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911, A320923, A320924.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],UnsameQ@@#&]!={}&]

A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

Original entry on oeis.org

1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616

Offset: 1

Gus Wiseman, Jan 09 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     27: {2,2,2}
     28: {1,1,4}
     30: {1,2,3}
     81: {2,2,2,2}
     84: {1,1,2,4}
     88: {1,1,1,5}
     90: {1,2,2,3}
    100: {1,1,3,3}
    208: {1,1,1,1,6}
    243: {2,2,2,2,2}
    252: {1,1,2,2,4}

Partitions of 2n into n parts are counted by A000041.
The number of prime indices alone is A001222.
The sum of prime indices alone is A056239.
Allowing sum to be any multiple of length gives A067538, ranked by A316413.
A000569 counts graphical partitions, ranked by A320922.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.
Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]

All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050

Offset: 0

Henry Bottomley, Aug 01 2005

nonn

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

			a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2.
a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6.
From _Gus Wiseman_, Oct 30 2018: (Start)
The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (2111)   (222)     (331)      (332)
               (1111)  (11111)  (321)     (2221)     (422)
                                (2211)    (3211)     (431)
                                (3111)    (22111)    (2222)
                                (21111)   (31111)    (3221)
                                (111111)  (211111)   (3311)
                                          (1111111)  (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (22)  (32)  (33)   (43)   (44)
            (31)  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (222)  (322)  (71)
                        (321)  (331)  (332)
                        (411)  (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (2222)
                                      (3221)
                                      (3311)
                                      (4211)
                                      (5111)
The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2},{1,2,3,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
(End)

Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921.

A000070 := proc(n) add( combinat[numbpart](i),i=0..n) ; end proc:
A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011

f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multhyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multhyp[#]!={}&]],{n,8}] (* Gus Wiseman, Oct 30 2018 *)

a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018

a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2).

a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018

A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 102, 342, 1213, 4361, 16016, 59348, 222117, 836315, 3166852, 12042620, 45967479, 176005709, 675759564, 2600672458, 10029832754, 38753710486, 149990133774, 581393603996, 2256710139346, 8770547818956, 34125389919850, 132919443189544, 518232001761434, 2022337118015338, 7898574056034636, 30873421455729728

Offset: 0

N. J. A. Sloane

In other words, a(n) is the number of graphic sequences of length n, where a graphic sequence is a sequence of numbers which can be the degree sequence of some graph.

In the article by A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, "Parallel enumeration of degree sequences of simple graphs II", in Table 4 on page 260 the values a(30) = 7898574056034638 and a(31) = 30873429530206738 are incorrect due to the incorrect Gz(30) = 5876236938019300 and Gz(31) = 22974847474172100. - Wang Kai, Jun 05 2016

			For n = 3, there are 4 different graphic sequences possible: 0 0 0; 1 1 0; 2 1 1; 2 2 2. - Daan van Berkel (daan.v.berkel.1980(AT)gmail.com), Jun 25 2010
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(4) = 11 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)  (0,0,0,0)
           (1,1)  (0,1,1)  (0,0,1,1)
                  (1,1,2)  (0,1,1,2)
                  (2,2,2)  (0,2,2,2)
                           (1,1,1,1)
                           (1,1,1,3)
                           (1,1,2,2)
                           (1,2,2,3)
                           (2,2,2,2)
                           (2,2,3,3)
                           (3,3,3,3)
For example, the graph {{2,3},{2,4}} has degrees (0,2,1,1), so (0,1,1,2) is counted under a(4).
(End)

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (Terms 1 through 31 were computed by various authors; terms 32 through 34 by Axel Kohnert and Wang Kai; terms 35 to 79 by Wang Kai)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995), #R11.
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
D. Dimitrov, Efficient computation of trees with minimal atom-bond connectivity index, arXiv:1305.1155 [cs.DM], 2013.
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapientiae, Inform. 3(2) (2011), 230-268.
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012), 260-288. - From _N. J. A. Sloane_, Feb 15 2013
A. Ivanyi and J. E. Schoenfield, Deciding football sequences, Acta Univ. Sapientiae, Informatica, 4, 1 (2012), 130-183. - From _N. J. A. Sloane_, Dec 22 2012 [Disclaimer: I am not one of the authors of this paper; I was unpleasantly surprised to find my name on it, as explained here. - _Jon E. Schoenfield_, Nov 26 2016]
Wang Kai, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms.
P. W. Mills, R. P. Rundle, V. M. Dwyer, T. Tilma, and S. J. Devitt, A proposal for an efficient quantum algorithm solving the graph isomorphism problem, arXiv 1711.09842, 2017.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Degree Sequence.
Eric Weisstein's World of Mathematics, Graphic Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
Index entries for sequences related to graphical partitions

Counting the positive partitions by sum gives A000569, ranked by A320922.
The version with half-loops is A029889, with covering case A339843.
The covering case (no zeros) is A095268.
Covering simple graphs are ranked by A309356 and A320458.
Non-graphical partitions are counted by A339617 and ranked by A339618.
The version with loops is A339844, with covering case A339845.
A006125 counts simple graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320921 counts connected graphical partitions.
A322353 counts factorizations into distinct semiprimes.
A339659 counts graphical partitions of 2n into k parts.
A339661 counts factorizations into distinct squarefree semiprimes.
Cf. A004250, A007717, A181819, A320894, A320923, A339560, A339561, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{2}]]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

G.f. = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 102*x^6 + 342*x^7 + 1213*x^8 + ...

a(n) ~ c * 4^n / n^(3/4) for some constant c > 0. Computational estimates suggest c ≈ 0.099094. - Tom Johnston, Jan 18 2023

More terms from Torsten Sillke, torsten.sillke(AT)lhsystems.com, using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
a(19) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 19 2007
a(20)-a(23) from Nathann Cohen, Jul 09 2011
a(24)-a(29) from Antal Iványi, Nov 15 2011
a(30) and a(31) corrected by Wang Kai, Jun 05 2016

A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 5, 8, 8, 13, 17, 22, 28, 39, 48, 62, 81, 101, 127, 167, 202, 253, 318, 395, 486, 608, 736, 906, 1113, 1353, 1637, 2011, 2409, 2922, 3510, 4227, 5060, 6089, 7242, 8661, 10306, 12251, 14503, 17236, 20345, 24045, 28334, 33374, 39223, 46076

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

Naturally, such a partition must have an even number of parts. Its multiplicities form a graphical partition (A000569, A320922), and vice versa.

			The a(3) = 1 through a(11) = 13 partitions (A = 10):
  (21)  (31)  (32)  (42)  (43)    (53)    (54)    (64)    (65)
              (41)  (51)  (52)    (62)    (63)    (73)    (74)
                          (61)    (71)    (72)    (82)    (83)
                          (3211)  (3221)  (81)    (91)    (92)
                                  (4211)  (3321)  (4321)  (A1)
                                          (4221)  (5221)  (4322)
                                          (4311)  (5311)  (4331)
                                          (5211)  (6211)  (4421)
                                                          (5321)
                                                          (5411)
                                                          (6221)
                                                          (6311)
                                                          (7211)
For example, the partition y = (4,3,3,2,1,1) can be partitioned into a set of edges in two ways:
  {{1,2},{1,3},{3,4}}
  {{1,3},{1,4},{2,3}},
so y is counted under a(14).

Eric Weisstein's World of Mathematics, Graphical partition.

A338916 allows equal pairs (x,x).
A339559 counts the complement in even-length partitions.
A339561 gives the Heinz numbers of these partitions.
A339619 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
A339659 counts graphical partitions of 2n into k parts.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A320893, A338899, A338903, A339564.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],strs[Times@@Prime/@#]!={}&]],{n,0,15}]

A027187(n) = a(n) + A339559(n).

More terms from Jinyuan Wang, Feb 14 2025

A339561 Products of distinct squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166

Offset: 1

Gus Wiseman, Dec 13 2020

nonn

First differs from A320911 in lacking 36.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      1: {}        55: {3,5}         91: {4,6}
      6: {1,2}     57: {2,8}         93: {2,11}
     10: {1,3}     58: {1,10}        94: {1,15}
     14: {1,4}     60: {1,1,2,3}     95: {3,8}
     15: {2,3}     62: {1,11}       106: {1,16}
     21: {2,4}     65: {3,6}        111: {2,12}
     22: {1,5}     69: {2,9}        115: {3,9}
     26: {1,6}     74: {1,12}       118: {1,17}
     33: {2,5}     77: {4,5}        119: {4,7}
     34: {1,7}     82: {1,13}       122: {1,18}
     35: {3,4}     84: {1,1,2,4}    123: {2,13}
     38: {1,8}     85: {3,7}        126: {1,2,2,4}
     39: {2,6}     86: {1,14}       129: {2,14}
     46: {1,9}     87: {2,10}       132: {1,1,2,5}
     51: {2,7}     90: {1,2,2,3}    133: {4,8}
For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
  (6*10*15*77)
  (6*10*21*55)
  (6*10*33*35)
  (6*14*15*55)
  (6*15*22*35)
  (10*14*15*33)
  (10*15*21*22),
so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A309356 is a kind of universal embedding.
A320894 is the complement in A028260.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A339560 counts the partitions with these Heinz numbers.
A339661 has nonzero terms at these positions.
A001358 lists semiprimes, with squarefree case A006881.
A005117 lists squarefree numbers.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqs[#]!={}&]

A339561 \/ A320894 = A028260.

A320921 Number of connected graphical partitions of 2n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 19, 35, 60

Offset: 0

Gus Wiseman, Oct 24 2018

An integer partition is connected and graphical if it comprises the multiset of vertex-degrees of some connected simple graph.

			The a(1) = 1 through a(6) = 19 connected graphical partitions:
  (11)  (211)  (222)   (2222)   (3322)    (3333)
               (2211)  (3221)   (22222)   (33222)
               (3111)  (22211)  (32221)   (33321)
                       (32111)  (33211)   (42222)
                       (41111)  (42211)   (43221)
                                (222211)  (222222)
                                (322111)  (322221)
                                (331111)  (332211)
                                (421111)  (333111)
                                (511111)  (422211)
                                          (432111)
                                          (522111)
                                          (2222211)
                                          (3222111)
                                          (3321111)
                                          (4221111)
                                          (4311111)
                                          (5211111)
                                          (6111111)

Gus Wiseman, Connected simple graphs realizing each of the a(6) = 19 connected graphical partitions of 12.

Cf. A000070, A000569, A007717, A025065, A096373, A147878, A209816, A320891, A320911, A320923.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[strnorm[2*n],Select[prptns[#],And[UnsameQ@@#,Length[csm[#]]==1]&]!={}&]],{n,5}]

A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 23, 29, 39, 53, 69, 90, 118, 150, 195, 249, 315, 398, 506, 629, 789, 982, 1219, 1504, 1860, 2277, 2798, 3413, 4161, 5051, 6137, 7406, 8948, 10765, 12943, 15503, 18571, 22153, 26432, 31432, 37352, 44268, 52444, 61944, 73141

Offset: 0

Gus Wiseman, Dec 09 2020

nonn

These are also integer partitions that can be partitioned into not necessarily distinct edges (pairs of distinct parts). For example, (3,3,2,2) can be partitioned as {{2,3},{2,3}}, so is counted under a(10), but (4,2,2,2) and (4,2,1,1,1,1) cannot be partitioned into edges. The multiplicities of such a partition form a multigraphical partition (A209816, A320924).

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
              (41)  (51)    (52)    (62)    (63)      (73)
                    (2211)  (61)    (71)    (72)      (82)
                            (3211)  (3221)  (81)      (91)
                                    (3311)  (3321)    (3322)
                                    (4211)  (4221)    (4321)
                                            (4311)    (4411)
                                            (5211)    (5221)
                                            (222111)  (5311)
                                                      (6211)
                                                      (322111)

Eric Weisstein's World of Mathematics, Graphical partition.

A096373 counts the complement in even-length partitions.
A320911 gives the Heinz numbers of these partitions.
A339560 is the strict case.
A339562 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320656 counts factorizations into squarefree semiprimes.
A320921 counts connected graphical partitions, ranked by A320923.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A005117, A007717, A030229, A320655, A322353, A338899, A338903.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]

A027187(n) = a(n) + A096373(n).

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A320924 Heinz numbers of multigraphical partitions.

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A004250 Number of partitions of n into 3 or more parts.

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A320922 Heinz numbers of graphical partitions.

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A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

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A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

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A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

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A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

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