TOPICS
Search

p-Good Path


A lattice path from one point to another is p-good if it lies completely below the line

 y=(p-1)x.
(1)

Hilton and Pederson (1991) show that the number of p-good paths from (1, q-1) to (k, n-k) under the condition 2<=k<=n-p+1<=p(k-1) is

 (n-q; k-1)-sum_(j=1)^l_pd_(qj)(n-pj; k-j),
(2)

where (a; b) is a binomial coefficient, and

 l=|_(n-k)/(p-1)_|,
(3)

where |_x_| is the floor function.


See also

Catalan Number, Lattice Path, Schröder Number

Explore with Wolfram|Alpha

References

Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13, 64-75, 1991.

Referenced on Wolfram|Alpha

p-Good Path

Cite this as:

Weisstein, Eric W. "p-Good Path." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/p-GoodPath.html

Subject classifications