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A321728
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Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.
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10
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0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50
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OFFSET
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0,5
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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.
Conjecture: a(n) is the number of non-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 zero, where m is monomial and e is elementary symmetric functions.
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EXAMPLE
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The a(2) = 1 through a(9) = 14 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 non-half-loop-graphical partitions up to n = 9:
(2) (3) (4) (5) (6) (7) (8) (9)
(31) (32) (33) (43) (44) (54)
(41) (42) (52) (53) (63)
(51) (61) (62) (72)
(411) (331) (71) (81)
(421) (422) (432)
(511) (431) (441)
(521) (522)
(611) (531)
(5111) (621)
(711)
(4311)
(5211)
(6111)
For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
{{1,1},{1,2},{1,3},{2,2}}
{{1},{2},{1,1},{1,2},{2,3}}
{{1},{2},{1,1},{1,3},{2,2}}
{{1},{3},{1,1},{1,2},{2,2}}
None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).
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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], Select[spsu[ptnverts[#], ptnpos[#]], Function[p, Sort[Length/@p]==Sort[#]]]=={}&]], {n, 8}]
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CROSSREFS
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The complement is counted by A321729.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339655.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A339659 counts graphical partitions of 2n into k parts.
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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