Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting pairs of lattice paths by intersections   On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin
at the SW corner, proceed with unit steps in either of the directions E or N,
and terminate at the NE corner of the rectangle. For each integer $k$ we ask
for $N_k^{n,r}$, the number of {\em ordered\/} pairs of these walks that
intersect in exactly $k$ points. The number of points in the intersection of
two such walks is defined as the cardinality of the intersection of their two
sets of vertices, excluding the initial and terminal vertices. We find two
explicit formulas for the numbers $N_k^{n,r}$. Next we note that $N_1^{n,r}= 2
N_0^{n,r}$, i.e., that {\em exactly twice as many pairs of walks have a single
intersection as have no intersection\/}. Such a relationship clearly merits a
bijective proof, and we supply one. We discuss a number of related results for
different assumptions on the two walks. We find the probability that two
independent walkers on a given lattice rectangle do not meet. In this
situation, the walkers start at the two points $(a,b+x+1)$ and (a+x+1,b)$ in
the first quadrant, and walk West or South at each step, except that when a
walker reaches the $x$-axis (resp. the $y$-axis) then all future steps are
constrained to be South (resp. West) until the origin is reached. We find that
if the probability $p(i,j)$ that a step from $(i,j)$ will go West depends only
on $i+j$, then the probabilty that the two walkers do not meet until they reach
the origin is the same as the probability that a single (unconstrained) walker
who starts at $(a, b+x+1)$ and and takes $a+b+x$ steps, finishes at one of the
points $(0,1), (-1,2), \ldots, (-x,1+x)$.

0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg