A330217 -id:A330217 - cp's OEIS frontend

A330234 Number of achiral factorizations of n into factors > 1.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 0, 1, 2, 2, 5, 1, 0, 1, 0, 2, 2, 1, 0, 2, 2, 3, 0, 1, 2, 1, 7, 2, 2, 2, 5, 1, 2, 2, 0, 1, 2, 1, 0, 0, 2, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 11, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 2, 1, 0, 5, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

A multiset of multisets is achiral if it is not changed by any permutation of the vertices. A factorization is achiral if taking the multiset of prime indices of each factor gives an achiral multiset of multisets.

			The a(n) factorizations for n = 2, 6, 27, 36, 243, 216:
  (2)  (6)    (27)     (36)       (243)        (216)
       (2*3)  (3*9)    (4*9)      (3*81)       (6*36)
              (3*3*3)  (6*6)      (9*27)       (8*27)
                       (2*3*6)    (3*9*9)      (12*18)
                       (2*2*3*3)  (3*3*27)     (4*6*9)
                                  (3*3*3*9)    (6*6*6)
                                  (3*3*3*3*3)  (2*3*36)
                                               (2*3*4*9)
                                               (2*3*6*6)
                                               (2*2*3*3*6)
                                               (2*2*2*3*3*3)

The fully chiral version is A330235.
Planted achiral trees are A003238.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
MM-numbers of achiral multisets of multisets are A330232.
Cf. A001055, A007716, A112798, A317533, A330098, A330227, A330228, A330236.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==1&]],{n,100}]

A330224 Number of achiral integer partitions of n.

Original entry on oeis.org

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

Gus Wiseman, Dec 08 2019

nonn

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.

			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)

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.

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}]

More terms from Jinyuan Wang, Jun 26 2020

A330218 Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.

Original entry on oeis.org

0, 5, 12, 180, 35636, 13

Offset: 1

Gus Wiseman, Dec 09 2019

nonn

A set-system is a finite set of finite nonempty sets of positive integers.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of set-systems together with their BII-numbers begins:
      0: {}
      5: {{1},{1,2}}
     12: {{1,2},{3}}
    180: {{1,2},{1,3},{2,3},{4}}
  35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
     13: {{1},{1,2},{3}}

Positions of first appearances in A330231.
The MM-number version is A330230.
Achiral set-systems are counted by A083323.
BII-numbers of fully chiral set-systems are A330226.
Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330098, A330101, A330195, A330217, A330229, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[bpe/@bpe[n]]],{n,0,1000}];
Table[Position[dv,i][[1,1]]-1,{i,First[Split[Union[dv],#1+1==#2&]]}]

cp's OEIS Frontend

A330234 Number of achiral factorizations of n into factors > 1.

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A330224 Number of achiral integer partitions of n.

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A330218 Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.

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