A330224 Number of achiral integer partitions of n.

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 30, 32, 43, 46, 57, 64, 79, 83, 103, 107, 130, 141, 162, 171, 205, 214, 245, 258, 297, 307, 357, 373, 423, 441, 493, 513, 591, 607, 674, 702, 790, 817, 917, 938, 1040, 1078, 1186, 1216, 1362, 1395, 1534, 1580, 1738, 1779, 1956

Offset

0,3

Comments

A multiset of multisets is achiral if it is not changed by any permutation of the vertices. An integer partition is achiral if taking the multiset of prime indices of each part gives an achiral multiset of multisets.

Links

Table of n, a(n) for n=0..54.

Example

The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (52)
             (111)  (31)    (41)     (42)      (61)
                    (211)   (221)    (51)      (331)
                    (1111)  (311)    (222)     (421)
                            (2111)   (321)     (511)
                            (11111)  (411)     (2221)
                                     (2211)    (3211)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[IntegerPartitions[n], Length[graprms[primeMS/@#]]==1&]], {n, 0, 30}]

Crossrefs

The fully-chiral version is A330228.

The Heinz numbers of these partitions are given by A330232.

Achiral set-systems are counted by A083323.

BII-numbers of achiral set-systems are A330217.

Non-isomorphic achiral multiset partitions are A330223.

Achiral factorizations are A330234.

Cf. A001055, A056239, A112798, A319564, A330098, A330230.

Sequence in context: A217147 A029732 A037950 * A322527 A181160 A316968

Adjacent sequences: A330221 A330222 A330223 * A330225 A330226 A330227

Keyword

nonn

Author

Gus Wiseman, Dec 08 2019

Extensions

More terms from Jinyuan Wang, Jun 26 2020

Status

approved