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.

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

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Comments

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

Examples

			The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

Links

Crossrefs

Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

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,IntegerPartitions[n]}],{n,14}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(12) 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

A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

Original entry on oeis.org

1, 1, 1, 4, 11, 51, 162, 876, 3761, 20782, 109293, 678569, 4038388, 27644436, 186524145, 1379760895, 10323844183, 82864869803, 674798169662, 5832742205056, 51385856585637, 474708148273586, 4486977535287371, 44152005855084345, 444577220573083896
Offset: 0

Views

Author

Gus Wiseman, Dec 16 2018

Keywords

Examples

			The a(4) = 11 set partitions:
  {{1},{2},{3},{4}}
   {{1},{2},{3,4}}
   {{1},{2,3},{4}}
   {{1},{2,4},{3}}
   {{1,2},{3},{4}}
   {{1,3},{2},{4}}
   {{1,4},{2},{3}}
    {{1},{2,3,4}}
    {{1,2,3},{4}}
    {{1,2,4},{3}}
    {{1,3,4},{2}}

Links

Crossrefs

a(n) >= A271426(n).
Cf. A000110, A000258, A000311, A000670, A000740, A000837, A005651, A005804, A008277, A124794, A319182, A320289.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t<2, 1, 0),
      add(b(n-j, igcd(t, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2019
  • Mathematica
    numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[Total[numSetPtnsOfType/@Select[IntegerPartitions[n],GCD@@#==1&]],{n,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, If[t < 2, 1, 0],
     Sum[b[n - j, GCD[t, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{|y| = n, GCD(y) = 1} n! / (Product_i y_i! * Product_i (y)_i!) where the sum is over all relatively prime integer partitions of n and (y)_i is the multiplicity of i in y.
Showing 1-4 of 4 results.

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