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%I #10 Jan 05 2021 21:34:18
%S 1,1,1,2,3,4,6,8,12,16,22,28,40,51
%N Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.
%C First differs from A046682 at a(11) = 28, A046682(11) = 29.
%C 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:
%C 1 2 3
%C 1 2
%C 2 3
%C 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.
%F a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is nonzero, where m is monomial symmetric functions and e is elementary symmetric functions.
%F a(n) = A000041(n) - A321728(n).
%e 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:
%e (1) (11) (21) (22) (221) (222) (322) (332)
%e (111) (211) (311) (321) (2221) (2222)
%e (1111) (2111) (2211) (3211) (3221)
%e (11111) (3111) (4111) (3311)
%e (21111) (22111) (4211)
%e (111111) (31111) (22211)
%e (211111) (32111)
%e (1111111) (41111)
%e (221111)
%e (311111)
%e (2111111)
%e (11111111)
%e For example, the half-loop-graphs
%e {{1},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2},{1,3}}
%e both have degrees y = (3,2,2), so y is counted under a(7).
%t spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
%t Table[Length[Select[IntegerPartitions[n],Length[Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]]>0&]],{n,8}]
%Y The complement is counted by A321728.
%Y Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
%Y The following pertain to the conjecture.
%Y Half-loop-graphical partitions by length are A029889 or A339843 (covering).
%Y The version for full loops is A339656.
%Y A027187 counts partitions of even length, ranked by A028260.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
%Y A339659 is a triangle counting graphical partitions by length.
%Y Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Nov 18 2018
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