generalization of Dyck Path: size K upward steps
$begingroup$
One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 unit up).
Dyck Paths from Wolfram Mathworld:
http://mathworld.wolfram.com/DyckPath.html
The number of Dyck Paths is given by the Catalan numbers.
My question is, if we generalize this so that the 2 allowed types of line segments are ($1$ unit to the right) and ($K$ units upward), now how many "K-Dyck paths" are possible?
Can we further generalize this to allow $K+1$ kinds of step types: ($1$ unit right) vs. (any of $1, 2, ldots, K$ units upward)?
combinatorics combinations combinatorial-geometry geometric-probability catalan-numbers
edited Jan 8 at 2:38
El borito
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
$endgroup$
add a comment |
$begingroup$
One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 unit up).
Dyck Paths from Wolfram Mathworld:
http://mathworld.wolfram.com/DyckPath.html
The number of Dyck Paths is given by the Catalan numbers.
My question is, if we generalize this so that the 2 allowed types of line segments are ($1$ unit to the right) and ($K$ units upward), now how many "K-Dyck paths" are possible?
Can we further generalize this to allow $K+1$ kinds of step types: ($1$ unit right) vs. (any of $1, 2, ldots, K$ units upward)?
combinatorics combinations combinatorial-geometry geometric-probability catalan-numbers
edited Jan 8 at 2:38
El borito
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
$endgroup$
$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55
add a comment |
$begingroup$
One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 unit up).
Dyck Paths from Wolfram Mathworld:
http://mathworld.wolfram.com/DyckPath.html
The number of Dyck Paths is given by the Catalan numbers.
My question is, if we generalize this so that the 2 allowed types of line segments are ($1$ unit to the right) and ($K$ units upward), now how many "K-Dyck paths" are possible?
Can we further generalize this to allow $K+1$ kinds of step types: ($1$ unit right) vs. (any of $1, 2, ldots, K$ units upward)?
combinatorics combinations combinatorial-geometry geometric-probability catalan-numbers
edited Jan 8 at 2:38
El borito
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
$endgroup$
One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 unit up).
Dyck Paths from Wolfram Mathworld:
http://mathworld.wolfram.com/DyckPath.html
The number of Dyck Paths is given by the Catalan numbers.
My question is, if we generalize this so that the 2 allowed types of line segments are ($1$ unit to the right) and ($K$ units upward), now how many "K-Dyck paths" are possible?
Can we further generalize this to allow $K+1$ kinds of step types: ($1$ unit right) vs. (any of $1, 2, ldots, K$ units upward)?
combinatorics combinations combinatorial-geometry geometric-probability catalan-numbers
combinatorics combinations combinatorial-geometry geometric-probability catalan-numbers
edited Jan 8 at 2:38
El borito
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
edited Jan 8 at 2:38
El borito
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
edited Jan 8 at 2:38
El borito
575216
edited Jan 8 at 2:38
El borito
575216
edited Jan 8 at 2:38
El borito
575216
575216
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
asked Jan 8 at 2:23
sambajetsonsambajetson
280211
280211
$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55
add a comment |
$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55
$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55
$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55
add a comment |
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$begingroup$
I know that there is a nice answer for paths which go one unit up or $k$ units right at each step. This is equivalent to paths which go up or right one step, staying at or above $y=kx$. See www-users.math.umn.edu/~reiner/Classes/… (you want the weak version at the end).
$endgroup$
– Mike Earnest
Jan 8 at 2:55