OFFSET
1,2
COMMENTS
a(n) is the number of climbable arrangements that exist for sets of n adjacent "broken ladders" with height n, where a broken ladder is an array of n steps with some number of the steps unusable, the rest usable; an arrangement is the configuration of the locations of the broken rung(s) on the n ladders of height n; and a climbable arrangement is a set of ladders such that with movement up, down, left, and right, there exists a path from the bottom to the top.
Also, a(n) is the sum of the coefficients of exact spanning probabilities in 2d lattices along the second dimension for an n X n square lattice.
LINKS
Stephan Mertens, Percolation.
Jeremy Rebenstock, Python notebook for calculating and visualizing a(n)
Jeremy Rebenstock and Thomas Ladouceur, Illustration for a(2) = 7
R. M. Ziff and M. E. J. Newman, Convergence of threshold estimates for two-dimensional percolation, arXiv:cond-mat/0203496 [cond-mat.stat-mech], 2002.
FORMULA
Upper limit: a(n) <= 2^(n^2). This is the total number of boards possible.
Lower limit: a(n) >= 2^(n-1)*a(n-1) climbable paths (board before it, with a completely unbroken ladder) and we break any arrangement of rungs on the new ladder.
EXAMPLE
x indicates a broken rung, - a functional rung.
.
|-| |-| |x| |-| |-| |x| |-| |-|
|-| |-| (1) |-| |-| (2) |-| |-| (3) |-| |x| (4)
.
|-| |-| |x| |-| |-| |x| |-| |-|
|x| |-| (5) |x| |-| (6) |-| |x| (7) |x| |x| (8)
.
|x| |x| |x| |-| |-| |x| |x| |x|
|-| |-| (9) |-| |x| (10) |x| |-| (11) |-| |x| (12)
.
|x| |x| |x| |-| |-| |x| |x| |x|
|x| |-| (13) |x| |x| (14) |x| |x| (15) |x| |x| (16)
.
The only climbable configurations are 1-7 since there is a path to the top from the bottom. So a(2) = 7.
PROG
(Python) see link
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Jeremy Rebenstock and Thomas Ladouceur, Sep 24 2023
STATUS
approved