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.

Showing 1-4 of 4 results.

A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

Original entry on oeis.org

1, 1, 3, 10, 61, 410, 3630
Offset: 0

Views

Author

Gus Wiseman, Dec 25 2019

Keywords

Comments

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 10 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}

Crossrefs

The version with multisets as leaves is A330465.
The singleton-reduced case is A330626.
A labeled version is A330625 (strongly normal).
The case with all atoms distinct is A141268.
The case where all leaves are singletons is A330470.
Cf. A000669, A004111, A005804, A007716, A273873, A300660, A320296, A330628, A330668, A330677.

A320294 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 37, 48, 87, 126, 227, 342, 611, 964, 1719, 2806, 4975, 8327, 14782, 25157, 44609, 76972, 136622, 237987, 422881, 742149, 1320825, 2331491, 4156392, 7370868, 13164429, 23433637, 41928557, 74871434, 134203411, 240284935, 431437069
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Comments

Also phylogenetic trees with no singleton leaves on integer partitions of n with no 1's.

Examples

			The a(4) = 1 through a(10) = 15 trees:
  (22)  (32)  (33)   (43)   (44)        (54)        (55)
              (42)   (52)   (53)        (63)        (64)
              (222)  (322)  (62)        (72)        (73)
                            (332)       (333)       (82)
                            (422)       (432)       (433)
                            (2222)      (522)       (442)
                            ((22)(22))  (3222)      (532)
                                        ((22)(23))  (622)
                                                    (3322)
                                                    (4222)
                                                    (22222)
                                                    ((22)(24))
                                                    ((22)(33))
                                                    ((23)(23))
                                                    ((22)(222))

Links

Crossrefs

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A304966, A304967, A319312, A320289, A320293, A320295, A320296.

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,10}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=2, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 25 2018

A320293 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n with no 1's.

Original entry on oeis.org

0, 1, 1, 3, 3, 9, 11, 30, 45, 112, 195, 475, 901, 2136, 4349, 10156, 21565, 50003, 109325, 252761, 563785, 1303296, 2948555, 6826494, 15604053, 36210591, 83415487, 194094257, 449813607, 1049555795, 2444027917, 5718195984, 13367881473, 31357008065, 73546933115
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Comments

Also phylogenetic trees on integer partitions of n with no 1's.

Examples

			The a(2) = 1 through a(7) = 11 trees:
  (2)  (3)  (4)       (5)       (6)            (7)
            (22)      (32)      (33)           (43)
            ((2)(2))  ((2)(3))  (42)           (52)
                                (222)          (322)
                                ((2)(4))       ((2)(5))
                                ((3)(3))       ((3)(4))
                                ((2)(22))      ((2)(23))
                                ((2)(2)(2))    ((3)(22))
                                ((2)((2)(2)))  ((2)(2)(3))
                                               ((2)((2)(3)))
                                               ((3)((2)(2)))

Links

Crossrefs

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A300660, A304967, A319312, A320289, A320294, A320295, A320296.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(23) and beyond from Andrew Howroyd, Oct 25 2018

A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890
Offset: 0

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Examples

			The a(4) = 1 through a(10) = 15 multiset partitions:
  ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
                  ((33))   ((34))   ((35))      ((36))      ((37))
                  ((222))  ((223))  ((44))      ((45))      ((46))
                                    ((224))     ((225))     ((55))
                                    ((233))     ((234))     ((226))
                                    ((2222))    ((333))     ((235))
                                    ((22)(22))  ((2223))    ((244))
                                                ((22)(23))  ((334))
                                                            ((2224))
                                                            ((2233))
                                                            ((22222))
                                                            ((22)(24))
                                                            ((22)(33))
                                                            ((23)(23))
                                                            ((22)(222))

Links

Crossrefs

Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

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]]]];
Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018

Formula

Euler transform of A083751. - Andrew Howroyd, Oct 25 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 25 2018
Showing 1-4 of 4 results.

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