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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..54.

Eric Weisstein's World of Mathematics, Degree Sequence.

Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Formula

Equals A320924 /\ A339618.

Equals A320924 \ A320922.

Example

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.

Mathematica

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[#]]!={}&]

Crossrefs

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.

Sequence in context: A020252 A076486 A270337 * A068529 A096059 A326742

Adjacent sequences: A339839 A339840 A339841 * A339843 A339844 A339845

Keyword

nonn

Author

Gus Wiseman, Dec 27 2020

Status

approved