A339845 Number of distinct sorted degree sequences among all n-vertex loop-graphs without isolated vertices.

1, 1, 4, 10, 35, 111, 392, 1364, 4925, 17845, 65654, 242966, 906417

Offset: 0

Author

Gus Wiseman, Dec 27 2020

Keywords

• nonn
• more

Comments

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). 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.

The following are equivalent characteristics for any positive integer n:

(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;

(2) n can be factored into distinct semiprimes;

(3) the prime signature of n is loop-graphical.

Examples

			The a(0) = 1 through a(3) = 10 sorted degree sequences:
  ()  (2)  (1,1)  (1,1,2)
           (1,3)  (1,1,4)
           (2,2)  (1,2,3)
           (3,3)  (1,3,4)
                  (2,2,2)
                  (2,2,4)
                  (2,3,3)
                  (2,4,4)
                  (3,3,4)
                  (4,4,4)
For example, the loop-graphs
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{3,3},{1,3}}
  {{1,1},{2,2},{3,3},{2,3}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{3,3},{1,2},{2,3}}
  {{2,2},{3,3},{1,2},{1,3}}
all have degrees y = (3,3,2), so y is counted under a(3).
		

Links

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

Crossrefs

See link for additional cross references.

The version without loops is A004251, with covering case A095268.

The half-loop version is A029889, with covering case A339843.

Loop-graphs are counted by A322661 and ranked by A320461 and A340020.

Counting the same partitions by sum gives A339656.

These partitions are ranked by A339658.

The non-covering case (zeros allowed) is A339844.

A007717 counts unlabeled multiset partitions into pairs.

A027187 counts partitions of even length, ranked by A028260.

A058696 counts partitions of even numbers, ranked by A300061.

A101048 counts partitions into semiprimes.

A339655 counts non-loop-graphical partitions of 2n.

A339659 counts graphical partitions of 2n into k parts.

Cf. A001358, A006125, A006129, A062740, A338898, A339841.

Programs

Mathematica

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{1,2}]/.{x_Integer}:>{x,x}],Union@@#==Range[n]&]]],{n,0,5}]

Formula

a(n) = A339844(n) - A339844(n-1) for n > 0. - Andrew Howroyd, Jan 10 2024

Extensions

a(7)-a(12) from Andrew Howroyd, Jan 10 2024