A349408 -id:A349408 - cp's OEIS frontend

A349409 Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with 0 <= k < n leaf-matched pairs. A leaf matched pair is a pair of non-leaf vertices (u,v) in the tanglegram such that the induced subtrees rooted and u and v also form a tanglegram (equivalently, the leaves in these two subtrees are matched by the matching that forms the original tanglegram).

1, 0, 1, 0, 1, 1, 0, 5, 4, 2, 0, 34, 28, 11, 3, 0, 273, 239, 102, 29, 6, 0, 2436, 2283, 1045, 325, 73, 11, 0, 23391, 23475, 11539, 3852, 968, 181, 23, 0, 237090, 254309, 133690, 47640, 12923, 2756, 444, 46, 0, 2505228, 2864283, 1605280, 607743, 175976, 40903, 7650, 1085, 98

Offset: 1

Kevin Liu, Nov 16 2021

The generating function can be proven using a generalization of the proof for A349408.

			Triangle begins
      1;
      0,      1;
      0,      1,     1;
      0,      5,     4,     2;
      0,     34,    28,    11,    3;
      0,    273,   239,   102,   29,   6;
      0,   2436,  2283,  1045,  325,  73,  11;
      0,  23391, 23475, 11539, 3852, 968, 181, 23;
      ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Kevin Liu, Planar Tanglegram Layouts and Single Edge Insertion, Séminaire Lotharingien de Combinatoire (2022) Vol. 86, Issue B, Art. No. 45.

Cf. A257887 (2nd column), A349408 (row sums), A001190 (diagonal).

\\ 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}
F(n)={my(h=H(n-2), p=O(x)); for(n=1, n, p = x + y*subst(h + O(x*x^n), x, p) + y*(p^2 + subst(subst(p,x,x^2),y,y^2))/2); p}
T(n)={[Vecrev(p) | p<-Vec(F(n))]}
{my(v=T(10)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Nov 18 2021

G.f.: F(x,q) = q*H(F(x,q)) + x + q*(F(x,q)^2 + F(x^2,q^2))/2 where coefficient of x^n*q^k is the number of planar tanglegrams with size n and k leaf-matched pairs, and H(x)/x^2 is the g.f. for A257887.

A371659 Triangle read by rows: T(n,k) is the number of planar tanglegrams of size n with irreducible component of size k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 5, 0, 13, 9, 20, 34, 0, 90, 46, 70, 170, 273, 0, 747, 312, 360, 680, 1638, 2436, 0, 7040, 2580, 2435, 3570, 7371, 17052, 23391, 0, 71736, 24056, 19800, 23970, 39858, 85260, 187128, 237090, 0, 774738, 243483, 182850, 193664, 267813, 477456, 1029204, 2133810, 2505228

Offset: 1

Kevin Liu, Apr 01 2024

A proper subtanglegram of a planar tanglegram is a pair of subtrees whose leaves are matched in the tanglegram, and the irreducible component of a planar tanglegram is formed by contracting each maximal proper subtanglegram into a pair of matched leaves.

			Triangle begins
  1;
  0,    1;
  0,    1,    1;
  0,    3,    3,    5;
  0,   13,    9,   20,   34;
  0,   90,   46,   70,  170,  273;
  0,  747,  312,  360,  680, 1638,  2436;
  0, 7040, 2580, 2435, 3570, 7371, 17052, 23391;
  ...

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, Advances in Applied Mathematics 149 (2023), Paper No. 102550.

Cf. A349408 (diagonal), A257887 (row sums).

G.f.: F(x,y) = H(F(x),y) + x*y + y^2*(F(x)^2 + F(x^2))/2 where the coefficient of x^n*y^k is the number of planar tanglegrams of size n with irreducible component of size k, F(x) is the g.f. for A349408, and H(x)/x^2 is the g.f. for A257887.

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