OFFSET

0,2

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.

LINKS

Eric Weisstein's World of Mathematics, Degree Sequence.

EXAMPLE

The a(0) = 1 through a(3) = 16 sorted degree sequences:

() (0) (0,0) (0,0,0)

(2) (0,2) (0,0,2)

(1,1) (0,1,1)

(1,3) (0,1,3)

(2,2) (0,2,2)

(3,3) (0,3,3)

(1,1,2)

(1,1,4)

(1,2,3)

(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).

MATHEMATICA

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

CROSSREFS

See link for additional cross references.

The covering case (no zeros) is A339845.

A007717 counts unlabeled multiset partitions into pairs.

A101048 counts partitions into semiprimes.

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

A339656 counts loop-graphical partitions of 2n.

A339659 counts graphical partitions of 2n into k parts.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 27 2020

EXTENSIONS

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

STATUS

approved