A339645 -id:A339645 - cp's OEIS frontend

A318226 Number of inequivalent leaf-colorings of rooted trees with n nodes.

1, 1, 3, 8, 25, 80, 286, 1070, 4280, 17946, 78907, 361248, 1718001, 8456130, 42980034, 225066289, 1212028798, 6701265897, 37986122037, 220477639797, 1308833637621, 7938564964369, 49151551028767, 310388888456536, 1997635594602629, 13093695854320203, 87349973125826943

Offset: 1

Gus Wiseman, Aug 21 2018

nonn

			Inequivalent representatives of the a(5) = 25 leaf-colorings:
(1111) (11(1)) (1(11)) ((111)) ((1)(1)) (1((1))) ((1(1))) (((11))) ((((1))))
(1112) (11(2)) (1(12)) ((112)) ((1)(2)) (1((2))) ((1(2))) (((12)))
(1122) (12(1)) (1(22)) ((123))
(1123) (12(3)) (1(23))
(1234)

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.

Cf. A318227, A318228, A318229, A318230, A318231, A318234, A339645.

undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Rule@@@Table[{(undats[m])[[i]],i},{i,Length[undats[m]]}]],First[Sort[expnorm[m,1]]]]];expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#>=aft&]}]},Union@@(expnorm[#,aft+1]&/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]])]];
urt[n_]:=urt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[urt/@c]],{c,IntegerPartitions[n-1]}]];
slip[e_,l_,q_]:=ReplacePart[e,Rule@@@Transpose[{Position[e,l],q}]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Join@@Table[Union[expnorm/@Table[slip[tree,{},seq],{seq,Join@@Permutations/@allnorm[Count[tree,{},{0,Infinity},Heads->True]]}]],{tree,urt[n]}]],{n,7}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(Z=x*sv(1), p = Z + O(x^2)); for(n=2, n, p = Z-x + x*sEulerT(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(9) and beyond from Andrew Howroyd, Dec 10 2020

A318228 Number of inequivalent leaf-colorings of planted achiral trees with n nodes.

Original entry on oeis.org

1, 1, 3, 6, 13, 20, 43, 58, 115, 171, 323, 379, 1034, 1135, 2321, 4327, 8915, 9212, 33939, 34429, 128414, 234017, 417721, 418976, 2931624, 5096391, 11770830, 20357876, 64853630, 64858195

Offset: 1

Gus Wiseman, Aug 21 2018

In a planted achiral tree, all branches directly under any given branch are identical.

			Inequivalent representatives of the a(5) = 13 leaf-colorings:
  (1111)  ((111))  ((1)(1))  (((11)))  ((((1))))
  (1112)  ((112))  ((1)(2))  (((12)))
  (1122)  ((123))
  (1123)
  (1234)

Cf. A000081, A001190, A001678, A003238, A004111, A214577, A290689, A304486.

Cf. A318226, A318227, A318229, A318230, A318231, A318234, A339645.

\\ See links in A339645 for combinatorial species functions.
G(v)={my(t=2, p=sv(1)); for(i=1, #v, my(d=v[i]); if(d>1, p=sApplyCI(symGroupCycleIndex(d), d, p, t)); t=t*d+1); p}
cycleIndex(n)={my(recurse(r,v)=if(r==1, G(v), sumdiv(r-1, d, self()((r-1)/d, concat(d,v))))); recurse(n,[])}
a(n)={StructsByCycleIndex(n, cycleIndex(n), n)} \\ Andrew Howroyd, Dec 13 2020

a(9)-a(30) from Andrew Howroyd, Dec 11 2020

A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

Original entry on oeis.org

1, 1, 1, 2, 5, 18, 98, 779, 10589, 255790, 11633297, 1004417286, 163944008107, 50324877640599, 29001521193534445, 31396727025729968365, 63969154112074956299242, 245871360738448777028919520, 1787330701747389106609369225312, 24636017249593067184544456944967278

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint
Gus Wiseman, The a(5) = 18 simple connected graphs whose bridges are all leaves.

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322394, A322395.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
bridgelessGraphs(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); sSolve( gc + gcr^2/2 - sRaise(gcr,2)/2, x*sv(1)*sExp(gcr) )}
cycleIndexSeries(n)={1+sSubstOp(bridgelessGraphs(n), symGroupSeries(n))}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 31 2020

a(6)-a(10) from Andrew Howroyd, Dec 08 2018

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2020

A007145 Number of rooted bridgeless graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 4, 24, 193, 2420, 47912, 1600524, 93253226, 9694177479, 1822463625183, 625829508087155, 395785845695978077, 464137111800208818956, 1015091598240432264958267, 4160447480034069826186309689, 32088552194861245127627790541334

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..40 (terms 1..22 from R. J. Mathar, terms 21..22 corrected by Andrew Howroyd)
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305.

Cf. A004115, A007146.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(g=graphsSeries(n), gcr=sPoint(g)/g); sSolve( gcr, x*sv(1)*sExp(gcr) )}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 27 2020

Reference gives first 22 terms (terms a(21) and a(22) contain typos).

More terms from R. J. Mathar, Jun 06 2007

A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 6, 19, 14, 113, 30, 584, 1150, 4023, 112, 119866, 202, 432061, 5442765, 16646712, 594, 738090160, 980, 13160013662, 113864783987, 39049423043, 2510, 44452496723053, 19373518220009, 21970704599961, 8858890258339122, 43233899006497146, 9130, 4019875470540832643

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(4) = 19 multiset partitions:
  {{1,1,1,1}}, {{1,1},{1,1}}, {{1},{1},{1},{1}},
  {{1,1,1,2}}, {{1,1},{1,2}}, {{1},{1},{1},{2}},
  {{1,1,2,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1},{1},{2},{2}},
  {{1,1,2,3}}, {{1,1},{2,3}}, {{1,2},{1,3}}, {{1},{1},{2},{3}},
  {{1,2,3,4}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}, {{1},{2},{3},{4}}.

Cf. A000005, A000041, A007716, A038041, A255906, A298422, A306017, A306018, A306019, A306020, A306021, A317583.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],SameQ@@Length/@#&]],{n,6}]

\\ See links in A339645 for combinatorial species functions.
cycleIndex(n)={sum(n=1, n, x^n*sumdiv(n, d, sApplyCI(symGroupCycleIndex(d), d, symGroupCycleIndex(n/d), n/d))) + O(x*x^n)}
StronglyNormalLabelingsSeq(cycleIndex(15)) \\ Andrew Howroyd, Jan 01 2021

a(p) = 2*A000041(p) for prime p. - Andrew Howroyd, Jan 01 2021

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2021

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

A259115 Number of unrooted binary ordered tanglegrams of size n.

Original entry on oeis.org

1, 1, 1, 2, 4, 31, 243, 3532, 62810, 1390718, 36080361, 1076477512, 36281518847, 1363869480379, 56587508558171, 2569141702825037, 126714642738385906, 6747643861563535720, 385875940575529343271, 23588199955061659841248, 1535037278334227258123709, 105961521687913311720698169

Offset: 1

Frederick A. Matsen IV, Jun 18 2015

nonn

Binary tanglegrams are pairs of bifurcating (degree 3 internal node) trees with a bijection between the leaves of the trees. Two tanglegrams are isomorphic if there is an isomorphism between the trees that preserves the bijection. Unrooted means that the tanglegram is composed of unrooted trees, and ordered means that the trees are considered as an ordered pair.

Andrew Howroyd, Table of n, a(n) for n = 1..50
S. C. Billey, M. Konvalinka, and F. A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.
Ira M. Gessel, Counting tanglegrams with species, arXiv:1509.03867 [math.CO], (13-September-2015)
F. A. Matsen IV, S. C. Billey, D. A. Kas, and M. Konvalinka, Tanglegrams: a reduction tool for mathematical phylogenetics, arXiv:1507.04784 [q-bio.PE], 2015.
Frederick A. Matsen, Sage/GAP4 Code for generating tanglegrams

Cf. A258620 (tanglegrams), A259114, A259116, A258486 (tangled chains), A258487, A258488, A258489.

\\ See links in A339645 for combinatorial species functions.
rootedBinTrees(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, n, v[n]=(sum(j=1, n-1, v[j]*v[n-j]) + if(n%2, 0, sRaiseCI(v[n/2], n/2, 2)))/2); x*Ser(v)}
cycleIndexSeries(n)={my(g=rootedBinTrees(n), u = g + (sRaise(g,3) - g^3)/3); sCartProd(u,u)}
NumUnlabeledObjsSeq(cycleIndexSeries(12)) \\ Andrew Howroyd, Dec 24 2020

More terms from Ira M. Gessel, Jul 19 2015

Terms a(15) and beyond from Andrew Howroyd, Dec 24 2020

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

Original entry on oeis.org

1, 0, 1, 0, 4, 2, 0, 10, 8, 3, 0, 33, 48, 18, 5, 0, 91, 204, 118, 32, 7, 0, 298, 959, 743, 266, 58, 11, 0, 910, 4193, 4334, 1927, 519, 94, 15, 0, 3017, 18947, 25305, 13992, 4407, 966, 154, 22, 0, 9945, 84798, 145033, 97947, 36410, 9023, 1679, 236, 30

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
   1
   0   1
   0   4   2
   0  10   8   3
   0  33  48  18   5
   0  91 204 118  32   7
   0 298 959 743 266  58  11
For example, row n = 3 counts the following multiset partitions:
  {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
  {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
  {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
  {{1}{11}}    {{2}}{{11}}
  {{1}{12}}    {{1}}{{1}{1}}
  {{1}{23}}    {{1}}{{1}{2}}
  {{2}{11}}    {{1}}{{2}{3}}
  {{1}{1}{1}}  {{2}}{{1}{1}}
  {{1}{1}{2}}
  {{1}{2}{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A007716 (for n > 0).
Column k = n is A000041.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
If this is the 3-dimensional version, the 2-dimensional version is A317533.
See A330473 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k,n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A,k,x)*x^k + O(x*x^n), sExp(A)) ))}
M(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 17 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2023

A340024 Number of inequivalent vertex colorings of graphs on n unlabeled vertices.

Original entry on oeis.org

1, 1, 4, 14, 89, 788, 13712, 459380, 31395800, 4304547500, 1170501781632, 626269787446920, 657129205489027200, 1350883625562244545584, 5441806297331472273603040, 42987375826579901036722653600, 666538741644051928632441002162384, 20306710978262167791045247702178986496

Offset: 0

Andrew Howroyd, Jan 01 2021

nonn

Equivalence is up to permutation of the colors. Adjacent vertices may have the same color.

Cf. A000088, A340022, A340023, A340027.

\\ See links in A339645 for combinatorial species functions.
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
InequivalentColoringsSeq(graphsSeries(15))

A340027 Number of inequivalent vertex colorings of connected graphs on n unlabeled vertices.

Original entry on oeis.org

1, 1, 2, 7, 50, 520, 10665, 400220, 29204589, 4143245857, 1146827743079, 619412332805088, 653237982066620540, 1346571060160843394520, 5432476352054378478159877, 42947950068987980977264834190, 666212968663987333085874313873428, 20301440661023158546856805172595805762

Offset: 0

Andrew Howroyd, Jan 02 2021

nonn

Equivalence is up to permutation of the colors. Adjacent vertices may have the same color.

			a(3) = 7 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be coloring in 3 ways (111, 112, 123) and the path graph P_3 can be colored in 4 ways (111, 112, 121, 123).

Cf. A340024, A340025, A340026.

\\ See links in A339645 for combinatorial species functions.
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
InequivalentColoringsSeq(1+sLog(graphsSeries(15)))

cp's OEIS Frontend

A318226 Number of inequivalent leaf-colorings of rooted trees with n nodes.

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A318228 Number of inequivalent leaf-colorings of planted achiral trees with n nodes.

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A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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A007145 Number of rooted bridgeless graphs with n nodes.

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A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

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A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

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A259115 Number of unrooted binary ordered tanglegrams of size n.

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PARI

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A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

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