login
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.
10

%I #13 Jan 05 2021 21:34:08

%S 0,0,1,1,2,3,5,7,10,14,20,28,37,50

%N 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 First differs from A000701 at a(11) = 28, A000701(11) = 27

%C A vertical section is a partial Young diagram with at most one square in each row.

%C 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>

%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>

%F 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 a(n) = A000041(n) - A321729(n).

%e 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 (2) (3) (4) (5) (6) (7) (8) (9)

%e (31) (32) (33) (43) (44) (54)

%e (41) (42) (52) (53) (63)

%e (51) (61) (62) (72)

%e (411) (331) (71) (81)

%e (421) (422) (432)

%e (511) (431) (441)

%e (521) (522)

%e (611) (531)

%e (5111) (621)

%e (711)

%e (4311)

%e (5211)

%e (6111)

%e For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:

%e {{1,1},{1,2},{1,3},{2,2}}

%e {{1},{2},{1,1},{1,2},{2,3}}

%e {{1},{2},{1,1},{1,3},{2,2}}

%e {{1},{3},{1,1},{1,2},{2,2}}

%e None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).

%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],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]

%Y The complement is counted by A321729.

%Y Cf. A000110, A000258, A000700, A000701, A008277, A046682, 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 A339655.

%Y A027187 counts partitions of even length, with Heinz numbers 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 counts graphical partitions of 2n into k parts.

%Y Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Nov 18 2018