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The ten monotone lattice paths from $(0, 0)$ to $(3, 2)$

Example

The ten monotone paths from $(0, 0)$ to $(3, 2)$ are exactly the length- $5$ words in ${E, N}$ with two $N$ steps. Grouped by the positions of the $N$ steps, they are:

positions of the $N$ steps	step word
${3, 4}$	`EEENN`
${2, 4}$	`EENEN`
${2, 3}$	`EENNE`
${1, 4}$	`ENEEN`
${1, 3}$	`ENENE`
${1, 2}$	`ENNEE`
${0, 4}$	`NEEEN`
${0, 3}$	`NEENE`
${0, 2}$	`NENEE`
${0, 1}$	`NNEEE`

Facts & Assumptions

Given: the step set ${E, N}$ .

[L1]

The number of monotone paths from $(0, 0)$ to $(m, n)$ is $(\binom{m + n}{n})$ , and the same translation formula holds for arbitrary endpoints ( $∣ M ((0, 0), (m, n)) ∣ = (\binom{m + n}{n})$ ).

Verification

technique · direct

1.1

Every word in the table has length $5$ with three $E$ steps and two $N$ steps, so each is a monotone path from $(0, 0)$ to $(3, 2)$ .

given

1.2

Every monotone path from $(0, 0)$ to $(3, 2)$ has length $5$ with exactly two $N$ steps, so its step word appears in the table at the row indexed by those two positions.

2.1

The table has ten rows, which agrees with [L1] because $(\binom{5}{2}) = 10$ .

L1step 1.1step 1.2

3.1

The boundary cases of the theorem are visible too: there is one path from $(0, 0)$ to $(3, 0)$ , namely EEE, and one path from $(0, 0)$ to $(0, 0)$ , namely the empty path.

L1∎

Remarks

The table is the concrete instance of the subset bijection used in the proof of the general counting theorem: the path is determined by the positions of its north steps.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Krattenthaler, "Lattice Path Enumeration", ch. 10 of the Handbook of Enumerative Combinatorics, §10.2 (standard reference, not scraped)

The ten monotone lattice paths from (0,0)(0,0) to (3,2)(3,2)