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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. 4
0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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

LINKS

Table of n, a(n) for n=0..13.

FORMULA

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.

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

EXAMPLE

The a(2) = 1 through a(9) = 14 partitions:

  (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)

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

CROSSREFS

Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321729, A321730, A321737, A321738.

Sequence in context: A027340 A000701 A123975 * A214077 A094984 A107332

Adjacent sequences:  A321725 A321726 A321727 * A321729 A321730 A321731

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 18 2018

STATUS

approved

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Last modified July 12 12:18 EDT 2020. Contains 335661 sequences. (Running on oeis4.)