A319312 -id:A319312 - cp's OEIS frontend

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

1, 1, 2, 4, 8, 17, 30, 61, 110, 207, 381, 711, 1250

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 17 multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,1,1}}    {{2,2}}        {{1,1,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,2,2}}
                  {{1},{1,1}}  {{1},{3}}      {{1},{4}}
                               {{1,1,1,1}}    {{2},{3}}
                               {{1},{1,2}}    {{1,1,1,2}}
                               {{2},{1,1}}    {{1},{1,3}}
                               {{1},{1,1,1}}  {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

A320174 Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 6, 19, 55, 200, 713, 2740, 10651, 42637, 173012, 713280, 2972389, 12514188, 53119400, 227140464, 977382586, 4229274235, 18391269922, 80330516578, 352269725526, 1550357247476, 6845517553493, 30316222112019, 134626183784975, 599341552234773, 2674393679352974

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(1) = 1 through a(4) = 19 trees:
  (1)  (2)       (3)            (4)
       (11)      (111)          (22)
       ((1)(1))  ((1)(2))       (1111)
                 ((1)(11))      ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((2)(11))
                                ((1)(111))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)(1)(11))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)((1)(11)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320175.

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]]]];
dot[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[dot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[dot[m],{m,IntegerPartitions[n]}]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 13, 37, 120, 395, 1381, 4931, 18074, 67287, 254387, 972559, 3756315, 14629237, 57395490, 226613217, 899773355, 3590349661, 14390323014, 57907783039, 233867667197, 947601928915, 3851054528838, 15693587686823, 64114744713845, 262543966114921, 1077406218930902

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(1) = 1 through a(4) = 13 trees:
  (1)  (2)       (3)            (4)
       ((1)(1))  (21)           (31)
                 ((1)(2))       ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((1)(21))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320174.

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]]]];
sot[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[sot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[sot[m],{m,IntegerPartitions[n]}]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,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) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 11, 29, 82, 247, 782, 2579, 8702, 29975, 104818, 371111, 1327307, 4788687, 17404838, 63669763, 234237605, 866090021, 3216738344, 11995470691, 44894977263, 168582174353, 634939697164, 2398004674911, 9079614633247, 34458722286825, 131059771522401

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(4) = 11 rooted identity trees:
  (1)  (2)   (3)        (4)
       (11)  (21)       (22)
             (111)      (31)
             ((1)(2))   (211)
             ((1)(11))  (1111)
                        ((1)(3))
                        ((1)(21))
                        ((2)(11))
                        ((1)(111))
                        ((1)((1)(2)))
                        ((1)((1)(11)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A319312, A320172, A320177, A320178.

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]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[gig[y]],{y,IntegerPartitions[n]}],{n,8}]

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]=numbpart(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(12) and beyond from Andrew Howroyd, Oct 25 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

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			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))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320178.

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

Terms a(13) and beyond from 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

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			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))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320177.

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

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A320289 Number of phylogenetic trees with n labels and no singleton leaves.

Original entry on oeis.org

0, 1, 1, 4, 11, 86, 477, 4810, 40679, 496522, 5662933, 81759910, 1169640551, 19622623190, 336215135973, 6455705990674, 128445712218263, 2785761076726066, 62980942321570981, 1525318051255683598, 38566041706375722071, 1032726237783455193662

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

			The a(2) = 1 through a(5) = 11 phylogenetic trees:
  (12)  (123)  (1234)      (12345)
               ((12)(34))  ((12)(345))
               ((13)(24))  ((13)(245))
               ((14)(23))  ((14)(235))
                           ((15)(234))
                           ((23)(145))
                           ((24)(135))
                           ((25)(134))
                           ((34)(125))
                           ((35)(124))
                           ((45)(123))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
rotf[n_]:=rotf[n]=If[n==1,0,1+Sum[numSetPtnsOfType[p]*Times@@rotf/@p,{p,Select[IntegerPartitions[n],Length[#]>1&]}]];
Array[rotf,20]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=2, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); 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

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

Gus Wiseman, Oct 09 2018

nonn

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

			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))

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

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}]

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

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

Gus Wiseman, Oct 09 2018

nonn

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

			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)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

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}]

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

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 6, 15, 22, 51, 86, 195, 354, 781, 1512, 3286, 6602, 14269, 29424, 63494, 133298, 287909, 612188, 1325375, 2844448, 6176145, 13348858, 29074164, 63187176, 138044144, 301350424, 660265471, 1446678326, 3178246273, 6985464590, 15384556290

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with n unlabeled objects and no singleton leaves.

			The a(2) = 1 through a(9) = 22 trees:
2   3   4     5     6        7        8           9
        (22)  (23)  (24)     (25)     (26)        (27)
                    (33)     (34)     (35)        (36)
                    (222)    (223)    (44)        (45)
                    (2(22))  ((22)3)  (224)       (225)
                             (2(23))  (233)       (234)
                                      (2222)      (333)
                                      ((22)4)     (2223)
                                      (2(24))     ((22)5)
                                      ((23)3)     (2(25))
                                      (2(33))     ((23)4)
                                      (2(222))    (2(34))
                                      (22(22))    ((24)3)
                                      ((22)(22))  ((33)3)
                                      (2(2(22)))  (2(22)3)
                                                  (2(223))
                                                  (22(23))
                                                  (3(222))
                                                  ((2(22))3)
                                                  ((22)(23))
                                                  (2((22)3))
                                                  (2(2(23)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A002865, A005804, A141268, A304967, A319312, A320293.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
t[n_]:=t[n]=If[PrimeQ[n],{n},Join@@Table[Union[Sort/@Tuples[t/@fac]],{fac,Select[facs[n],Length[#]>1&]}]];
Table[Sum[Length[t[Times@@Prime/@ptn]],{ptn,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,15}]

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

Terms a(26) and beyond from Andrew Howroyd, Oct 25 2018

cp's OEIS Frontend

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

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A320174 Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

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A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

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A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A320177 Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

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A320178 Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

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A320289 Number of phylogenetic trees with n labels and no singleton leaves.

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