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A321729
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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.
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8
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1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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MATHEMATICA
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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}]
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CROSSREFS
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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.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339656.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A339659 is a triangle counting graphical partitions by length.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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