A278679 Popularity of left children in treeshelves avoiding pattern T213.

1, 5, 24, 128, 770, 5190, 38864, 320704, 2894544, 28382800, 300575968, 3419882304, 41612735632, 539295974000, 7417120846080, 107904105986048, 1655634186628352, 26721851169634560, 452587550053179392, 8026445538106839040, 148751109541600495104

Offset: 2

Author

Sergey Kirgizov, Nov 26 2016

Keywords

• nonn

Comments

Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T213 illustrated below corresponds to a treeshelf constructed from permutation 213. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.

Examples

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T213:
    1
   / \
  2   \
       3
Treeshelves of size 3 that avoid pattern T213:
      1  1          1    1        1
     /    \        /      \      / \
    2      2      /        \    /   2
   /        \    2          2  3
  3          3    \        /
                   3      3
Popularity of left children is 5.
		

Links

• Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
• J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), p. 35-50.

Crossrefs

Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678.

Programs

Mathematica

terms = 21;
egf = (E^(Sqrt[2] z)(4z - 4) - (Sqrt[2] - 2) E^(2 Sqrt[2] z) + Sqrt[2] + 2)/((Sqrt[2] - 2) E^(Sqrt[2] z) + 2 + Sqrt[2])^2;
CoefficientList[egf + O[z]^(terms + 2), z]*Range[0, terms + 1]! // Round // Drop[#, 2]& (* Jean-François Alcover, Jan 26 2019 *)

Python

## by Taylor expansion
from sympy import *
from sympy.abc import z
h = (exp(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*exp(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*exp(sqrt(2)*z) + 2 + sqrt(2))**2
NUMBER_OF_COEFFS = 20
coeffs = Poly(series(h,n = NUMBER_OF_COEFFS)).coeffs()
coeffs.reverse()
## and remove first coefficient 1 that corresponds to O(n**k)
coeffs.pop(0)
print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])

Formula

E.g.f.: (e^(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*e^(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*e^(sqrt(2)*z) + 2 + sqrt(2))^2.

Asymptotic: n * (sqrt(2) / log(2*sqrt(2)+3) )^(n+1).

Extensions

More terms from Alois P. Heinz, Oct 27 2017