A258620 -id:A258620 - cp's OEIS frontend

A258487 Number of tangled chains of length k=4.

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.

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.

A258489 Number of tangled chains of length k=6.

Original entry on oeis.org

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.

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

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.

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

A349408 Number of planar tanglegrams of size n.

Original entry on oeis.org

1, 1, 2, 11, 76, 649, 6173, 63429, 688898, 7808246, 91537482, 1102931565, 13594564857, 170804438005, 2181426973452, 28257128116954, 370581034530685, 4913238656392058, 65773613137623085, 888155942037325535, 12086555915234897267, 165641209243876120135

Offset: 1

Kevin Liu, Nov 16 2021

nonn

			For n=4, there are 11 planar tanglegrams of size 4.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana and Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.

Row sums of A349409.

Cf. A257887, A258620.

\\ here H(n)/x^2 is g.f. of A257887.
H(n)={(x - x^2 - serreverse(sum(k=0, n+1, (binomial(2*k, k)/(k+1))^2*x^(k+1)) + O(x^(n+3))))/2}
seq(n)={my(h=H(n-2), p=O(x)); for(n=1, n, p = subst(h + O(x*x^n), x, p) + x + (p^2 + subst(p,x,x^2))/2); Vec(p)} \\ Andrew Howroyd, Nov 18 2021

G.f.: F(x) satisfies F(x) = H(F(x)) + x + (F(x)^2 + F(x^2))/2 where H(x)/x^2 is the g.f. of A257887.

Terms a(11) and beyond from Andrew Howroyd, Nov 18 2021

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

A339234 Number of series-reduced tanglegrams with n unlabeled leaves.

Original entry on oeis.org

1, 1, 5, 51, 757, 16416, 461231, 16021550, 662197510, 31749450007, 1732478051823, 106025572201434, 7192665669790893, 535756912504764218, 43471544417828923777, 3816784803681841133512, 360546156617986177328681, 36462349359125513109697520, 3930704977357944446111295571

Offset: 1

Andrew Howroyd, Jan 01 2021

nonn

A tanglegram is a pair of trees with their leaves superimposed. The original tanglegram sequence (A258620) used rooted binary trees. This variation uses planted series-reduced trees.

			Two of the 5 tanglegrams for a(3) are illustrated (A,B are the roots of the trees and o marks the leaves that are shared between the two trees)
           A             A
          /  \          /  \
         /   / \       /   / \
        o   o   o     o   o   o
         \  |  /       \ /   /
          \ | /          \  /
            B              B

Cf. A000669 (series-reduced trees), A258620 (binary tanglegrams), A339645.

\\ See links in A339645 for combinatorial species functions.
seriesReducedTrees(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
NumUnlabeledObjsSeq(sCartPower(seriesReducedTrees(15), 2))

cp's OEIS Frontend

A258487 Number of tangled chains of length k=4.

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A258488 Number of tangled chains of length k=5.

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A258489 Number of tangled chains of length k=6.

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A259114 Number of rooted binary unordered tanglegrams of size n.

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A258485 Number of tangled chains of length k=7.

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A259115 Number of unrooted binary ordered tanglegrams of size n.

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A349408 Number of planar tanglegrams of size n.

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A259116 Number of unrooted binary unordered tanglegrams of size n.

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A339234 Number of series-reduced tanglegrams with n unlabeled leaves.

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