A000569 -id:A000569 - cp's OEIS frontend

A339658 Heinz numbers of loop-graphical partitions (of even numbers).

1, 3, 4, 9, 10, 12, 16, 25, 27, 28, 30, 36, 40, 48, 63, 64, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 147, 160, 175, 189, 192, 196, 198, 208, 210, 220, 225, 243, 250, 252, 256, 264, 270, 280, 300, 324, 336, 343, 352, 360, 400, 432, 441, 448, 462, 468, 480

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320912.
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.
A partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms > 1 together with their prime indices begins:
      3: {2}               70: {1,3,4}          192: {1,1,1,1,1,1,2}
      4: {1,1}             75: {2,3,3}          196: {1,1,4,4}
      9: {2,2}             81: {2,2,2,2}        198: {1,2,2,5}
     10: {1,3}             84: {1,1,2,4}        208: {1,1,1,1,6}
     12: {1,1,2}           88: {1,1,1,5}        210: {1,2,3,4}
     16: {1,1,1,1}         90: {1,2,2,3}        220: {1,1,3,5}
     25: {3,3}            100: {1,1,3,3}        225: {2,2,3,3}
     27: {2,2,2}          108: {1,1,2,2,2}      243: {2,2,2,2,2}
     28: {1,1,4}          112: {1,1,1,1,4}      250: {1,3,3,3}
     30: {1,2,3}          120: {1,1,1,2,3}      252: {1,1,2,2,4}
     36: {1,1,2,2}        144: {1,1,1,1,2,2}    256: {1,1,1,1,1,1,1,1}
     40: {1,1,1,3}        147: {2,4,4}          264: {1,1,1,2,5}
     48: {1,1,1,1,2}      160: {1,1,1,1,1,3}    270: {1,2,2,2,3}
     63: {2,2,4}          175: {3,3,4}          280: {1,1,1,3,4}
     64: {1,1,1,1,1,1}    189: {2,2,2,4}        300: {1,1,2,3,3}
For example, the four loop-graphs with degrees y = (3,1,1,1) are:
  {{1,1},{1,2},{3,4}}
  {{1,1},{1,3},{2,4}}
  {{1,1},{1,4},{2,3}}
  {{1,2},{1,3},{1,4}},
so the Heinz number 40 is in the sequence. On the other hand, the three loop-multigraphs with degrees y = (4,4) are
  {{1,1},{1,1},{2,2},{2,2}}
  {{1,1},{1,2},{1,2},{2,2}}
  {{1,2},{1,2},{1,2},{1,2}},
but none of these is a loop-graph, so the Heinz number 49 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 has these prime shadows (see A181819).
A339656 counts these partitions.
A339657 ranks the complement, counted by A339655.
A001358 lists semiprimes, with squarefree case A006881.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A338914 can be partitioned into strict pairs (A320911).
- A338916 can be partitioned into distinct pairs (A320912).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A320892, A338898, A338912, A338913, A339112.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[25],Select[mpsbin[nrmptn[#]],UnsameQ@@#&]!={}&]

A300061 = A339657 \/ A339658.

A029889 Number of graphical partitions (degree-vectors for graphs with n vertices, allowing self-loops which count as degree 1; or possible ordered row-sum vectors for a symmetric 0-1 matrix).

Original entry on oeis.org

1, 2, 5, 14, 43, 140, 476, 1664, 5939, 21518, 78876, 291784, 1087441, 4077662, 15369327, 58184110, 221104527, 842990294, 3223339023

Offset: 0

torsten.sillke(AT)lhsystems.com

I call loops of degree one half-loops, so these are half-loop-graphs or graphs with half-loops. - Gus Wiseman, Dec 31 2020

			From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(3) = 14 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)
      (1)  (1,0)  (1,0,0)
           (1,1)  (1,1,0)
           (2,1)  (2,1,0)
           (2,2)  (2,2,0)
                  (1,1,1)
                  (2,1,1)
                  (3,1,1)
                  (2,2,1)
                  (3,2,1)
                  (2,2,2)
                  (3,2,2)
                  (3,3,2)
                  (3,3,3)
For example, the half-loop-graph
  {{1,3},{3}}
has degrees (1,0,2), so (2,1,0) is counted under a(3). The half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees (3,2,2), so (3,2,2) is counted under a(3).
(End)

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).
Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A029890, A029891.
Non-half-loop-graphical partitions are conjectured to be counted by A321728.
The covering case (no zeros) is A339843.
MM-numbers of half-loop-graphs are given by A340018 and A340019.
A004251 counts degree sequences of graphs, with covering case A095268.
A320663 counts unlabeled multiset partitions into singletons/pairs.
A339659 is a triangle counting graphical partitions.
A339844 counts degree sequences of loop-graphs, with covering case A339845.
Cf. A006125, A006129, A027187, A028260, A062740, A096373, A322661, A339560.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{1,2}]]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

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

a(n) = A029890(n) + A029891(n). - Andrew Howroyd, Apr 18 2021

a(0) = 1 prepended by Gus Wiseman, Dec 31 2020

A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

Original entry on oeis.org

1, 2, 4, 11, 33, 146, 939, 10932

Offset: 1

Sam Heil and Caleb Ji, Oct 04 2016

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895). - Gus Wiseman, Nov 21 2018

			For n = 3, under the p basis, the CSF's are: p_{1, 1, 1}, p_{1, 1, 1} - p_{2, 1}, p_{1, 1, 1} - 2p_{2, 1} + p_{3}, p_{1, 1, 1} - 3p_{2, 1} + 2p_{3}.
From _Gus Wiseman_, Nov 21 2018: (Start)
The a(4) = 11 chromatic symmetric functions (m is the augmented monomial symmetric function basis):
                                     m(1111)
                            m(211) + m(1111)
                           2m(211) + m(1111)
          m(22) +          2m(211) + m(1111)
                           3m(211) + m(1111)
          m(22) +          3m(211) + m(1111)
                   m(31) + 3m(211) + m(1111)
         2m(22) +          4m(211) + m(1111)
          m(22) +  m(31) + 4m(211) + m(1111)
         2m(22) + 2m(31) + 5m(211) + m(1111)
  m(4) + 3m(22) + 4m(31) + 6m(211) + m(1111)
(End)

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.

Cf. A000088, A000110, A000569, A006125, A229048, A240936, A245883, A277204, A277205, A321750, A321751, A321895, A321911.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
Table[Length[Union[chromSF/@simpleSpans[n]]],{n,6}] (* Gus Wiseman, Nov 21 2018 *)

A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

Original entry on oeis.org

7, 13, 19, 21, 22, 29, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 154, 155, 156, 159, 163, 165, 166, 169, 171

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320892.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms together with their prime indices begins:
      7: {4}         57: {2,8}      107: {28}
     13: {6}         61: {18}       111: {2,12}
     19: {8}         62: {1,11}     113: {30}
     21: {2,4}       66: {1,2,5}    115: {3,9}
     22: {1,5}       71: {20}       116: {1,1,10}
     29: {10}        76: {1,1,8}    117: {2,2,6}
     34: {1,7}       79: {22}       118: {1,17}
     37: {12}        82: {1,13}     121: {5,5}
     39: {2,6}       85: {3,7}      129: {2,14}
     43: {14}        87: {2,10}     130: {1,3,6}
     46: {1,9}       89: {24}       131: {32}
     49: {4,4}       91: {4,6}      133: {4,8}
     52: {1,1,6}     94: {1,15}     134: {1,19}
     53: {16}       101: {26}       136: {1,1,1,7}
     55: {3,5}      102: {1,2,7}    138: {1,2,9}
For example, the three loop-multigraphs with degrees y = (5,2,1) are:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320892 has these prime shadows (see A181819).
A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
A339655 counts these partitions.
A339658 ranks the complement, counted by A339656.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339844 counts loop-graphical partitions by length.
factorizations of n into distinct primes or 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).
- A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).
- 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).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[50],EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]],UnsameQ@@#&]=={}&]

A300061 = A339657 \/ A339658.

A339620 Heinz numbers of non-multigraphical partitions of even numbers.

Original entry on oeis.org

3, 7, 10, 13, 19, 21, 22, 28, 29, 34, 37, 39, 43, 46, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 155, 156, 159, 163, 166, 171, 172, 173

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);
(2) n can be factored into squarefree semiprimes;
(3) the unordered prime signature of n is multigraphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         53: {16}          94: {1,15}
      7: {4}         55: {3,5}        101: {26}
     10: {1,3}       57: {2,8}        102: {1,2,7}
     13: {6}         61: {18}         107: {28}
     19: {8}         62: {1,11}       111: {2,12}
     21: {2,4}       66: {1,2,5}      113: {30}
     22: {1,5}       71: {20}         115: {3,9}
     28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
     29: {10}        79: {22}         117: {2,2,6}
     34: {1,7}       82: {1,13}       118: {1,17}
     37: {12}        85: {3,7}        129: {2,14}
     39: {2,6}       87: {2,10}       130: {1,3,6}
     43: {14}        88: {1,1,1,5}    131: {32}
     46: {1,9}       89: {24}         133: {4,8}
     52: {1,1,6}     91: {4,6}        134: {1,19}
For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a multigraph (they have loops), the Heinz number 66 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A000070 counts these partitions.
A300061 is a superset.
A320891 has image under A181819 equal to this set of terms.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
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 [this sequence]).
- A209816 counts multigraphical partitions (A320924).
- A147878 counts connected multigraphical partitions (A320925).
- 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).
Cf. A001055, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.

prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&prpts[nrmptn[#]]=={}&]

Equals A300061 \ A320924.

For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.

A339659 Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 2, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 7, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 4, 9, 11, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 2, 11, 15, 17, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Gus Wiseman, Dec 18 2020

Conjecture: The column sums 1, 0, 1, 2, 7, 20, 67, ... are given by A304787.

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.

			Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 1 2 1 1
  0 0 0 0 2 3 2 1 1
  0 0 0 0 1 4 5 3 2 1 1
  0 0 0 0 1 4 7 7 5 3 2 1 1
For example, row n = 5 counts the following partitions:
  3322  22222  222211  2221111  22111111  211111111  1111111111
        32221  322111  3211111  31111111
        33211  331111  4111111
        42211  421111
               511111

A000569 gives the row sums.
A004250 is the central column.
A005408 gives the row lengths.
A008284/A072233 is the version counting all partitions.
A259873 is the left half of the triangle.
A309356 is a universal embedding.
A027187 counts partitions of even length.
A339559 = partitions that cannot be partitioned into distinct strict pairs.
A339560 = partitions that can be partitioned into distinct strict pairs.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A147878 counts connected multigraphical partitions (A320925).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000219, A002100, A006881, A007717, A025065, A320656, A320894, A338914, A338916, A339561, A339661.

prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Subsets[Union[m],{2}]}]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Length[Union[#]]==k&&Select[prpts[#],UnsameQ@@#&]!={}&]],{n,0,5},{k,0,2*n}]

A095268 Number of distinct degree sequences among all n-vertex graphs with no isolated vertices.

Original entry on oeis.org

1, 0, 1, 2, 7, 20, 71, 240, 871, 3148, 11655, 43332, 162769, 614198, 2330537, 8875768, 33924859, 130038230, 499753855, 1924912894, 7429160296, 28723877732, 111236423288, 431403470222, 1675316535350, 6513837679610, 25354842100894, 98794053269694, 385312558571890, 1504105116253904, 5876236938019298, 22974847399695092

Offset: 0

Eric W. Weisstein, May 31 2004

nonn

A002494 is the number of graphs on n nodes with no isolated points and A095268 is the number of these graphs having distinct degree sequences.
Now that more terms have been computed, we can see that this is not the self-convolution of any integer sequence. - Paul D. Hanna, Aug 18 2006
Is it true that a(n+1)/a(n) tends to 4? Is there a heuristic argument why this might be true? - Gordon F. Royle, Aug 29 2006
Previous values a(30) = 5876236938019300 from Lorand Lucz, Jul 07 2013 and a(31) = 22974847474172100 from Lorand Lucz, Sep 03 2013 are wrong. New values a(30) and a(31) independently computed Kai Wang and Axel Kohnert. - Vaclav Kotesovec, Apr 15 2016
In the article by A. Iványi, G. Gombos, L. Lucz, T. Matuszka: "Parallel enumeration of degree sequences of simple graphs II" is in the tables on pages 258 and 261 a wrong value a(31) = 22974847474172100, but in the abstract another wrong value a(31) = 22974847474172374. - Vaclav Kotesovec, Apr 15 2016
The asymptotic formula given below confirms that a(n+1)/a(n) tends to 4. - Tom Johnston, Jan 18 2023

			a(4) = 7 because a 4-vertex graph with no isolated vertices can have degree sequence 1111, 2211, 2222, 3111, 3221, 3322 or 3333.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(3) = 7 sorted degree sequences (empty column indicated by dot):
  ()  .  (1,1)  (2,1,1)  (1,1,1,1)
                (2,2,2)  (2,2,1,1)
                         (2,2,2,2)
                         (3,1,1,1)
                         (3,2,2,1)
                         (3,3,2,2)
                         (3,3,3,3)
For example, the complete graph K_4 has degrees y = (3,3,3,3), so y is counted under a(4). On the other hand, the only half-loop-graphs (up to isomorphism) with degrees y = (4,2,2,1) are: {(1),(1,2),(1,3),(1,4),(2,3)} and {(1),(2),(3),(1,2),(1,3),(1,4)}; and since neither of these is a graph (due to having half-loops), y is not counted under a(4).
(End)

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (terms 0 through 79 from Kai Wang)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
A. Iványi, L. Lucz, T. Matuszka and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapiantiae, Inform.4 (2) (2012) 260-288.
A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, Parallel enumeration of degree sequences of simple graphs II, Acta Universitatis Sapientiae, Informatica, Volume 5, Issue 2 (Dec 2013).
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapiantiae, Inform. 3 (2) (2011) 230-268.
A. Kohnert, Number of different degree sequences of a graph with no isolated vertices, 2016.
Frank Ruskey, Alley CATs in Search of Good Homes
Kai Wang, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms. But a(30) and a(31) are different.
Eric Weisstein's World of Mathematics, Degree sequence
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Cf. A002494, A004250, A007721 (analog for connected graphs), A271831.
Counting the same partitions by sum gives A000569.
Allowing isolated nodes gives A004251.
The version with half-loops is A029889, with covering case A339843.
Covering simple graphs are ranked by A309356 and A320458.
Graphical partitions are ranked by A320922.
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.
A339659 is a triangle counting graphical partitions.
Cf. A007717, A096373, A181819, A320921, A322661, A339559, A339560.

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

a(n) ~ c * 4^n / n^(3/4) for some c > 0. Computational estimates suggest c ≈ 0.074321. - Tom Johnston, Jan 18 2023

Edited by N. J. A. Sloane, Aug 26 2006
More terms from Gordon F. Royle, Aug 21 2006
a(21) and a(22) from Frank Ruskey, Aug 29 2006
a(23) from Frank Ruskey, Aug 31 2006
a(24)-a(29) from Matuszka Tamás, Jan 10 2013
a(30)-a(31) from articles by Kai Wang and Axel Kohnert, Apr 15 2016
a(0) = 1 and a(1) = 0 prepended by Gus Wiseman, Dec 31 2020

A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 3, 2, 2, 2, 0, 1, 3, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 4, 1, 0, 4

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

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 singletons or strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes.

			The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
  (6)    (5*6)    (6*10)    (6*35)     (2*6*35)
  (2*3)  (2*15)   (2*5*6)   (10*21)    (5*6*14)
         (3*10)   (2*3*10)  (14*15)    (6*7*10)
         (2*3*5)            (5*6*7)    (2*10*21)
                            (2*3*35)   (2*14*15)
                            (2*5*21)   (2*5*6*7)
                            (2*7*15)   (3*10*14)
                            (3*5*14)   (2*3*5*14)
                            (3*7*10)   (2*3*7*10)
                            (2*3*5*7)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008966 with A339661.
A008966 allows only primes.
A339661 does not allow primes, only squarefree semiprimes.
A339740 lists the positions of zeros.
A339741 lists the positions of positive terms.
A339839 allows nonsquarefree semiprimes.
A339887 is the non-strict version.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339840 cannot be factored into distinct primes or semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050320 into squarefree numbers.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339742 [this sequence] into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A058696 counts all partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A339656 counts loop-graphical partitions (A339658).
-
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 has no additional conditions (A028260).
- A338914/A339562 can be partitioned into edges (A320911).
- A338916/A339563 can be partitioned into distinct pairs (A320912).
- A339559/A339564 cannot be partitioned into distinct edges (A320894).
- A339560/A339619 can be partitioned into distinct edges (A339561).
Cf. A000070, A001221, A005117, A320893, A320923, A338899, A339113, A339617, A353471.

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

A353471(n) = (numdiv(n)==2*omega(n));
A339742(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA353471(d), s += A339742(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n squarefree} A339661(n/d).

More terms from Antti Karttunen, May 02 2022

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A339661 Number of factorizations of n into distinct squarefree semiprimes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Dec 19 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
                      (10*14*33)  (10*39*77)   (6*14*15*55)
                      (10*21*22)  (14*33*65)   (6*15*22*35)
                      (14*15*22)  (14*39*55)   (10*14*15*33)
                                  (15*22*91)   (10*15*21*22)
                                  (15*26*77)
                                  (21*22*65)
                                  (21*26*55)
                                  (22*35*39)
                                  (26*33*35)
                                  (6*35*143)
                                  (10*21*143)
                                  (14*15*143)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
A050326 allows all squarefree numbers, non-strict case A050320.
A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- 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:
- 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).
Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

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

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

More terms and secondary offset added by Antti Karttunen, May 02 2022

cp's OEIS Frontend

A339658 Heinz numbers of loop-graphical partitions (of even numbers).

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A029889 Number of graphical partitions (degree-vectors for graphs with n vertices, allowing self-loops which count as degree 1; or possible ordered row-sum vectors for a symmetric 0-1 matrix).

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A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

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A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

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A339620 Heinz numbers of non-multigraphical partitions of even numbers.

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A339659 Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.

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A095268 Number of distinct degree sequences among all n-vertex graphs with no isolated vertices.

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A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

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