OFFSET
0,4
COMMENTS
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
1 2 3
1 2
2 3
Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.
FORMULA
EXAMPLE
The a(1) = 1 through a(8) = 12 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the half-loop-graphical partitions up to n = 8:
(1) (11) (21) (22) (221) (222) (322) (332)
(111) (211) (311) (321) (2221) (2222)
(1111) (2111) (2211) (3211) (3221)
(11111) (3111) (4111) (3311)
(21111) (22111) (4211)
(111111) (31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
For example, the half-loop-graphs
{{1},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(7).
MATHEMATICA
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y], {k}], {k, Reverse[Union[y]]}], UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n], Length[Select[spsu[ptnverts[#], ptnpos[#]], Function[p, Sort[Length/@p]==Sort[#]]]]>0&]], {n, 8}]
CROSSREFS
The complement is counted by A321728.
Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
The version for full loops is A339656.
A339659 is a triangle counting graphical partitions by length.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 18 2018
STATUS
approved