Frederick A. Matsen IV - cp's OEIS frontend

A258489 Number of tangled chains of length k=6.

1, 1, 122, 474883, 11168414844, 989169269347359, 250335000079534559375, 151038989624520433840089358, 191158216491241179675824199407135, 461408865973380293005829125668717407727, 1973397409908124305318632313047269426852165625, 14104214451439837037643144221899175649593123932192274

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=6, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^6)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258488 Number of tangled chains of length k=5.

Original entry on oeis.org

1, 1, 41, 31732, 106420469, 1046976648840, 24085106680575625, 1117767454807330938472, 94308987414050519542935029, 13390317159105772877158700776107, 3014130596940522685213135526859317500, 1025828273466214412416440210115479183065903, 507888918625036626314714587415852381698509422634

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=5, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^5)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258485 Number of tangled chains of length k=7.

Original entry on oeis.org

1, 1, 365, 7119961, 1172597933594, 934741501255380321, 2602204282373953017437500, 20410544568790568555722851029455, 387481340785957748099474582410763014214, 15899856312608503503306403988460714538830399657

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=6, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^7)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258487 Number of tangled chains of length k=4.

Original entry on oeis.org

1, 1, 14, 2140, 1017219, 1110178602, 2320017306125, 8278981347401059, 46556715158334549170, 388779284837787599307987, 4605471565794120802036550000, 74633554055057890778698344509705, 1606481673354648219373898238155693682, 44821655543075499856527523557216582931002

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=4, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

S. Billey, Table of n, a(n) for n = 1..20
Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^4)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A259116 Number of unrooted binary unordered tanglegrams of size n.

Original entry on oeis.org

1, 1, 1, 2, 4, 22, 145, 1875, 31929, 698183, 18056523, 538340256, 18141423039, 681939320185

Offset: 1

Frederick A. Matsen IV, Jun 18 2015

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 unordered means that two tanglegrams that differ by exchanging the trees and inverting the bijection are considered identical.

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, A259115, A258486 (tangled chains), A258487, A258488, A258489.

More terms from Ira M. Gessel, Jul 19 2015

A259114 Number of rooted binary unordered tanglegrams of size n.

Original entry on oeis.org

1, 1, 2, 10, 69, 807, 13048, 269221, 6660455, 191411477, 6257905519, 229312906604, 9309547057292, 414803750101863

Offset: 1

Frederick A. Matsen IV, Jun 18 2015

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. Rooted means that the tanglegram is composed of rooted trees, and unordered means that two tanglegrams that differ by exchanging the trees and inverting the bijection are considered identical.

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), A259115, A259116, A258486 (tangled chains), A258487, A258488, A258489.

More terms from Ira M. Gessel, Jul 19 2015

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

cp's OEIS Frontend

User: Frederick A. Matsen IV

A258489 Number of tangled chains of length k=6.

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A258488 Number of tangled chains of length k=5.

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A258485 Number of tangled chains of length k=7.

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A258487 Number of tangled chains of length k=4.

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A259116 Number of unrooted binary unordered tanglegrams of size n.

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A259114 Number of rooted binary unordered tanglegrams of size n.

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A259115 Number of unrooted binary ordered tanglegrams of size n.

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