cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 22 results. Next

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689
Offset: 0

Views

Author

Gus Wiseman, Nov 02 2018

Keywords

Comments

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}

Crossrefs

Cf. A006126, A096827, A285572, A293993, A293994, A302545, A318099, A319560, A319719, A319721.
Cf. A320797, A320798, A320804, A320811, A320812, A320813.

A096826 Number of maximal-sized antichains in divisor lattice D(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 4, 3, 1, 2, 3, 2, 1, 1, 5, 2, 3, 2, 3, 1, 1, 2, 6, 3, 1, 4, 3, 2, 2, 2, 6, 1, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 3, 3, 1, 2, 10, 3, 3, 1, 3, 2, 6, 1, 6, 1, 1, 2, 1, 2, 1, 3, 7, 1, 2, 2, 3, 1, 2, 2, 4, 2, 1, 3, 3, 1, 2, 2, 10, 5, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

Keywords

Comments

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

Examples

			From _Gus Wiseman_, Aug 24 2018: (Start)
The a(120) = 6 antichains:
  {8,12,20,30}
  {8,12,15,20}
  {8,10,12,15}
  {6,8,15,20}
  {6,8,10,15}
  {4,6,10,15}
(End)

Links

Crossrefs

Cf. A000005, A008480, A096825, A096827, A253249, A285572 A285573.

Programs

  • Sage
    def A096826(n) :
    if n==1 : return 1
    R. = QQ[]; mults = [x[1] for x in factor(n)]
    maxsize = prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
    dlat = LatticePoset((divisors(n), attrcall("divides")))
    count = 0
    for ac in dlat.antichains_iterator() :
        if len(ac) == maxsize : count += 1
    return count
# Eric M. Schmidt, May 13 2013

Extensions

More terms from Eric M. Schmidt, May 13 2013

A318531 Number of finite sets of set partitions of {1,...,n} such that any two have join {{1,...,n}}.

Original entry on oeis.org

2, 4, 18, 450, 436270
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2018

Keywords

Examples

			The a(3) = 18 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,3},{2}}   {{1,2,3}}
   {{1,2},{3}}   {{1,2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1},{2,3}}   {{1,2,3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,2,3}}
   {{1,2},{3}}  {{1,3},{2}}   {{1,2,3}}
   {{1},{2,3}}  {{1,3},{2}}   {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}   {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}  {{1,2,3}}

Crossrefs

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
Cf. A318389, A318390, A318391, A318392, A318394, A318532.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[stableSets[sps[Range[n]],Length[csm[Union[#1,#2]]]>1&]],{n,4}]

A318532 Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}} and join {{1,...,n}}.

Original entry on oeis.org

2, 4, 11, 51, 635, 15591
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2018

Keywords

Examples

			The a(3) = 11 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,2},{3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,2},{3}}
  {{1},{2},{3}}   {{1,2,3}}
   {{1},{2,3}}   {{1,2},{3}}  {{1,3},{2}}

Crossrefs

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
Cf. A318389, A318390, A318391, A318392, A318394, A318531.

A321680 Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 3, 4, 9, 14, 39, 80, 216, 538, 1460
Offset: 0

Views

Author

Gus Wiseman, Nov 16 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset trees:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1},{1}}  {{1,2,3,3}}        {{1,2,2,3,3}}
                                   {{1,2,3,4}}        {{1,2,3,3,3}}
                                   {{1,1},{1,1}}      {{1,2,3,4,4}}
                                   {{1,2},{2,2}}      {{1,2,3,4,5}}
                                   {{1,3},{2,3}}      {{1,1},{1,2,2}}
                                   {{1},{1},{1},{1}}  {{1,2},{2,2,2}}
                                                      {{1,2},{2,3,3}}
                                                      {{1,3},{2,3,3}}
                                                      {{1,4},{2,3,4}}
                                                      {{3,3},{1,2,3}}
                                                      {{1},{1},{1},{1},{1}}

Crossrefs

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319719, A319721, A321194, A321585, A321681.

A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 77, 205, 517, 1399
Offset: 0

Views

Author

Gus Wiseman, Nov 16 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 13 trees:
  {{1}}  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
         {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
                  {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                             {{1,2,3,3}}    {{1,2,2,3,3}}
                             {{1,2,3,4}}    {{1,2,3,3,3}}
                             {{1,2},{2,2}}  {{1,2,3,4,4}}
                             {{1,3},{2,3}}  {{1,2,3,4,5}}
                                            {{1,1},{1,2,2}}
                                            {{1,2},{2,2,2}}
                                            {{1,2},{2,3,3}}
                                            {{1,3},{2,3,3}}
                                            {{1,4},{2,3,4}}
                                            {{3,3},{1,2,3}}

Crossrefs

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319557, A319565, A319719, A319721, A321585, A321680.

A326082 Number of maximal sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2019

Keywords

Comments

Depends only on prime signature.
The non-maximal case is A096827.

Examples

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Crossrefs

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

A175179 Primes for which value of CRT (Cardinality of rooted tree) is equal to 1.

Original entry on oeis.org

2, 3, 5, 11, 17, 19, 23, 29, 31, 47, 53, 67, 71, 79, 83, 89, 97, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 199
Offset: 1

Views

Author

Artur Jasinski, Mar 01 2010

Keywords

Comments

Primes p = Prime(x) such that A175178(x)=1.

References

  • Karpenko A.S. 2006. Lukasiewicz's Logics and Prime Numbers (English translation).
  • Karpenko A.S. 2000. Lukasiewicz's Logics and Prime Numbers (Russian).

Crossrefs

Cf. A096827, A175177, A175178.

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 7, 6, 15, 15, 30
Offset: 0

Views

Author

Gus Wiseman, Oct 29 2018

Keywords

Examples

			The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (211)   (11111)  (222)     (3211)     (332)
               (1111)           (321)     (22111)    (422)
                                (2211)    (31111)    (431)
                                (3111)    (211111)   (2222)
                                (21111)   (1111111)  (3221)
                                (111111)             (3311)
                                                     (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(6) = 7 integer partitions together with a realizing multi-antichain of each (the parts of the partition count the appearances of each vertex in the multi-antichain):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2,3},{1,2,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}

Crossrefs

Cf. A000070, A000569, A006126, A096827, A147878, A209816, A283877, A318360, A319719, A319721, A320799, A320921, A321176.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]

A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 11, 13, 39, 67, 174
Offset: 0

Views

Author

Gus Wiseman, Nov 16 2018

Keywords

Comments

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 11 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{3,4}}  {{1,2},{3,4,5}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{1,4},{2,3,4}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}

Crossrefs

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321679.
Previous Showing 11-20 of 22 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.