Consider the Lattice $\mathbb{Z}^2$ and an initial set of points with coordinates $(0,u_1)$, $(0,u_2)$, $\cdots$ $(0,u_n)$, final set of points $(m,v_1),(m,v_2),\cdots,(m,v_n)$, where $v_i,u_i$ are strictly increasing and $v_i\leq u_i$. The horizontal axis represents time.

I'm considering configurations of up-right paths starting at the points $u$ and ending at $v$ such that:

1) No two paths intersect or overlap at any given time

2) Each path can move only right or up one unit step at a given time

3) Only one path can go up at a given time $t$. Furthermore, after a particular path has taken a vertical step, it must take one horizontal step (so a path cannot go up two vertices at one time step).

Taken from wikipedia, here's a configuration (rotated 90 degrees, with time on the vertical axis) that's NOT allowed:

enter image description here

this is NOT allowed because the red path jumps twice at time 4.

I've seen literature on "vicious" walkers but none of them seem to satisfy condition 3. My set of paths bijects strictly to standard Young tableaux. Whereas vicious walkers biject to semi-standard tableaux.

Question: is there a general term for such walkers in the literature?


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