A320154
Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.
Original entry on oeis.org
1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680
Offset: 1
The a(1) = 1 through a(4) = 18 rooted trees:
(1) (12) (123) (1234)
((1)(2)) ((1)(23)) ((1)(234))
((2)(13)) ((12)(34))
((3)(12)) ((13)(24))
((1)(2)(3)) ((14)(23))
((2)(134))
((3)(124))
((4)(123))
((1)(2)(34))
((1)(3)(24))
((1)(4)(23))
((2)(3)(14))
((2)(4)(13))
((3)(4)(12))
((1)(2)(3)(4))
(((1)(2))((3)(4)))
(((1)(3))((2)(4)))
(((1)(4))((2)(3)))
Cf.
A000081,
A000311,
A000669,
A001678,
A005804,
A048816,
A079500,
A119262,
A120803,
A141268,
A244925,
A292504,
A300660,
A319312.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018
A320177
Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.
Original entry on oeis.org
1, 1, 3, 5, 11, 26, 65, 169, 463, 1294, 3691, 10700, 31417, 93175, 278805, 840424, 2549895, 7780472, 23860359, 73500838, 227330605, 705669634, 2197750615, 6865335389, 21505105039, 67533738479, 212575923471, 670572120240, 2119568530289, 6712115439347
Offset: 1
The a(1) = 1 through a(5) = 11 rooted trees:
(1) (2) (3) (4) (5)
(21) (31) (32)
((1)(2)) ((1)(3)) (41)
((1)(12)) ((1)(4))
((1)((1)(2))) ((2)(3))
((1)(13))
((2)(12))
((1)((1)(3)))
((2)((1)(2)))
((1)((1)(12)))
((1)((1)((1)(2))))
-
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]]]];
gog[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gog/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[gog[m],{m,IntegerPartitions[n]}]],{n,10}]
-
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018
A320178
Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
Original entry on oeis.org
1, 2, 4, 8, 19, 53, 151, 459, 1445, 4634, 15154, 50253, 168607, 571212, 1951588, 6715575, 23255444, 80978697, 283373024, 995995996, 3514614634, 12446666967, 44222390525, 157587392768, 563096832839, 2017121728223, 7242436444030, 26059512879605, 93952946906117
Offset: 1
The a(1) = 1 through a(5) = 19 rooted trees:
(1) (2) (3) (4) (5)
(11) (111) (22) (11111)
((1)(2)) (1111) ((1)(4))
((1)(11)) ((1)(3)) ((2)(3))
((2)(11)) ((1)(22))
((1)(111)) ((3)(11))
((1)((1)(2))) ((2)(111))
((1)((1)(11))) ((1)(1111))
((11)(111))
((1)(2)(11))
((1)((1)(3)))
((2)((1)(2)))
((11)((1)(2)))
((1)((2)(11)))
((2)((1)(11)))
((1)((1)(111)))
((11)((1)(11)))
((1)((1)((1)(2))))
((1)((1)((1)(11))))
-
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]]]];
gob[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gob/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[gob[m],{m,IntegerPartitions[n]}]],{n,10}]
-
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018
Showing 1-3 of 3 results.
Comments