

A321729


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.


8



1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51
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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 halfloopgraphical partitions of n. An integer partition is halfloopgraphical if it comprises the multiset of vertexdegrees of some graph with halfloops, where a halfloop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.


LINKS



FORMULA

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.


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 halfloopgraphical 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 halfloopgraphs
{{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.
Halfloopgraphical 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.


KEYWORD

nonn,more


AUTHOR



STATUS

approved



