A095268 -id:A095268 - cp's OEIS frontend

A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

9, 25, 30, 49, 63, 70, 75, 84, 100, 121, 147, 154, 165, 169, 175, 189, 196, 198, 210, 220, 250, 264, 273, 280, 286, 289, 325, 343, 351, 361, 363, 364, 385, 390, 441, 442, 462, 468, 484, 490, 495, 507, 520, 525, 529, 550, 561, 588, 594, 595, 616, 624, 637, 646

Offset: 1

Gus Wiseman, Dec 27 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph, and multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

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.

			The sequence of terms together with their prime indices begins:
      9: {2,2}        189: {2,2,2,4}      363: {2,5,5}
     25: {3,3}        196: {1,1,4,4}      364: {1,1,4,6}
     30: {1,2,3}      198: {1,2,2,5}      385: {3,4,5}
     49: {4,4}        210: {1,2,3,4}      390: {1,2,3,6}
     63: {2,2,4}      220: {1,1,3,5}      441: {2,2,4,4}
     70: {1,3,4}      250: {1,3,3,3}      442: {1,6,7}
     75: {2,3,3}      264: {1,1,1,2,5}    462: {1,2,4,5}
     84: {1,1,2,4}    273: {2,4,6}        468: {1,1,2,2,6}
    100: {1,1,3,3}    280: {1,1,1,3,4}    484: {1,1,5,5}
    121: {5,5}        286: {1,5,6}        490: {1,3,4,4}
    147: {2,4,4}      289: {7,7}          495: {2,2,3,5}
    154: {1,4,5}      325: {3,3,6}        507: {2,6,6}
    165: {2,3,5}      343: {4,4,4}        520: {1,1,1,3,6}
    169: {6,6}        351: {2,2,2,6}      525: {2,3,3,4}
    175: {3,3,4}      361: {8,8}          529: {9,9}
For example, a complete list of all multigraphs with degrees (4,2,2,2) is:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
Since none of these is strict, i.e., a graph, the Heinz number 189 is in the sequence.

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
Distinct prime shadows (images under A181819) of A340017.
A000070 counts non-multigraphical partitions (A339620).
A000569 counts graphical partitions (A320922).
A027187 counts partitions of even length (A028260).
A058696 counts partitions of even numbers (A300061).
A096373 cannot be partitioned into strict pairs.
A209816 counts multigraphical partitions (A320924).
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A320893 can be partitioned into distinct pairs but not into strict pairs.
A339560 can be partitioned into distinct strict pairs.
A339617 counts non-graphical partitions of 2n (A339618).
A339659 counts graphical partitions of 2n into k parts.
Cf. A004251, A006129, A007717, A056239, A095268, A112798, A181821, A305936, A318284, A339559.

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
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[#]]]&& Select[strr[Times@@Prime/@nrmptn[#]],UnsameQ@@#&]=={}&&strr[Times@@Prime/@nrmptn[#]]!={}&]

Equals A320924 /\ A339618.

Equals A320924 \ A320922.

A007721 Number of distinct degree sequences among all connected graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 6, 19, 68, 236, 863, 3137, 11636, 43306, 162728, 614142, 2330454, 8875656, 33924699, 130038017, 499753560, 1924912505, 7429159770, 28723877046, 111236422377, 431403469046, 1675316533812, 6513837677642, 25354842098354, 98794053266471, 385312558567775

Offset: 1

Frank Ruskey

Sometimes called "graphical partitions", although this term is deprecated.

Wang Kai, Table of n, a(n) for n = 1..118
N. Durand, G. Granger, A traffic complexity approach through cluster analysis, in proceedings of the 5th ATM R&D Seminar, Budapest, Hungary (2003)
T. Hoppe, A. Petrone,Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
F. Ruskey, Alley CATs in search of good homes, Congress. Numerant., 102 (1994) 97-110.
Kai Wang, Efficient Counting of Degree Sequences, arXiv preprint arXiv:1604.04148 [math.CO], 2017.
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A007722, A029889; A095268 (analog for all graphs).

a(9) corrected by Gordon Royle, Aug 30 2006
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 19 2007
Prepended missing term a(1), Travis Hoppe, Aug 04 2014
a(22)-a(28) added by Wang Kai, Feb 15 2017

A007722 Number of graphical partitions of biconnected graphs with n nodes.

Original entry on oeis.org

1, 3, 9, 34, 125, 473, 1779, 6732, 25492, 96927, 369463, 1412700, 5415117, 20807502, 80120350, 309106496, 1194609429, 4624160156, 17925278497, 69578272204, 270401326899, 1052036082719, 4097343156323, 15973179953261, 62325892264031, 243392644741599

Offset: 3

Frank Ruskey

nonn

F. Ruskey, Alley CATs in search of good homes, Congress. Numerant., 102 (1994) 97-110.

Wang Kai, Table of n, a(n) for n = 3..101
Kai Wang, Efficient Counting of Degree Sequences, arXiv preprint arXiv:1604.04148 [math.CO], 2016-2017.
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A007721, A029889, A095268.

a(15)-a(28) added by Kai Wang, Feb 15 2017

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A339842 Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.

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A007721 Number of distinct degree sequences among all connected graphs with n nodes.

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A007722 Number of graphical partitions of biconnected graphs with n nodes.

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