Delannoy number
Encyclopedia
In mathematics
, a Delannoy number
describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east.
For an n × n grid, the first few Delannoy numbers (starting with n=0) are :
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ...
The following figure illustrates the 63 Delannoy paths through a 3 × 3 grid:
The paths that do not rise above the SW–NE diagonal represent the Schröder number
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Delannoy number
describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east.For an n × n grid, the first few Delannoy numbers (starting with n=0) are :
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ...
The following figure illustrates the 63 Delannoy paths through a 3 × 3 grid:
The paths that do not rise above the SW–NE diagonal represent the Schröder number
Schröder number
In mathematics, a Schröder number describes the number of paths from the southwest corner of an n × n grid to the northeast corner , using only single steps north, northeast, or east, that do not rise above the SW–NE diagonal.The first few Schröder numbers are-Examples:The following...
s.