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%I A321728 #13 Jan 05 2021 21:34:08
%S A321728 0,0,1,1,2,3,5,7,10,14,20,28,37,50
%N A321728 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.
%C A321728 First differs from A000701 at a(11) = 28, A000701(11) = 27
%C A321728 A vertical section is a partial Young diagram with at most one square in each row.
%C A321728 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.
%H A321728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H A321728 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F A321728 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.
%F A321728 a(n) = A000041(n) - A321729(n).
%e A321728 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:
%e A321728 (2) (3) (4) (5) (6) (7) (8) (9)
%e A321728 (31) (32) (33) (43) (44) (54)
%e A321728 (41) (42) (52) (53) (63)
%e A321728 (51) (61) (62) (72)
%e A321728 (411) (331) (71) (81)
%e A321728 (421) (422) (432)
%e A321728 (511) (431) (441)
%e A321728 (521) (522)
%e A321728 (611) (531)
%e A321728 (5111) (621)
%e A321728 (711)
%e A321728 (4311)
%e A321728 (5211)
%e A321728 (6111)
%e A321728 For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
%e A321728 {{1,1},{1,2},{1,3},{2,2}}
%e A321728 {{1},{2},{1,1},{1,2},{2,3}}
%e A321728 {{1},{2},{1,1},{1,3},{2,2}}
%e A321728 {{1},{3},{1,1},{1,2},{2,2}}
%e A321728 None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).
%t A321728 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321728 ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t A321728 ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
%t A321728 Table[Length[Select[IntegerPartitions[n],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]
%Y A321728 The complement is counted by A321729.
%Y A321728 Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321730, A321737, A321738.
%Y A321728 The following pertain to the conjecture.
%Y A321728 Half-loop-graphical partitions by length are A029889 or A339843 (covering).
%Y A321728 The version for full loops is A339655.
%Y A321728 A027187 counts partitions of even length, with Heinz numbers A028260.
%Y A321728 A058696 counts partitions of even numbers, ranked by A300061.
%Y A321728 A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A321728 A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
%Y A321728 A339659 counts graphical partitions of 2n into k parts.
%Y A321728 Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.
%K A321728 nonn,more
%O A321728 0,5
%A A321728 _Gus Wiseman_, Nov 18 2018