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A two-by-two determinant counting non-intersecting path pairs
Example
At , the two monotone paths from to are EN and NE, and
the two monotone paths from to are again EN and NE.
Facts & Assumptions
Given: the four paths above.
The count of non-intersecting pairs is (For the pairs of non-intersecting monotone paths and number , The set of -element subsets and the binomial coefficient ).
Verification
The four ordered pairs of paths are (EN,EN), (EN,NE), (NE,EN) and (NE,NE).
Exactly one of them, (NE,EN), meets at the lattice point ; the other three are non-intersecting.
Therefore the direct count is , which matches [L1] because .
Remarks
- This is the smallest instance in which the determinant count differs from the product of the individual path counts.
Depends on
- For $n\ge1$ the pairs of non-intersecting monotone paths $(1,0)\to(n+1,n)$ and $(0,1)\to(n,n+1)$ number $\binom{2n}{n}^2-\binom{2n}{n+1}\binom{2n}{n-1}$
- $\det\bigl(\lvert\mathcal{M}(A_i,E_j)\rvert\bigr)_{i,j}=\sum_{\pi\in S_r}\operatorname{sgn}(\pi)\cdot\#\{\text{non-intersecting }\pi\text{-systems}\}$
- Path systems between two families of lattice points, and non-intersecting systems
- The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- C. Krattenthaler, "Lattice Path Enumeration", ch. 10 of the Handbook of Enumerative Combinatorics, Corollary 10.13.2 (standard reference, not scraped)