False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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FALSE: the monotone paths from $(0, 0)$ to $(n, n)$ staying weakly below the diagonal are exactly half of all monotone paths

Statement

False claim: among the monotone paths from $(0, 0)$ to $(n, n)$ , exactly half stay weakly below the diagonal $y = x$ .

Facts & Assumptions

Given: the case $n = 2$ .

[L1]

Replacing $U$ by $N$ and $D$ by $E$ gives a bijection in which diagonal height is $y - x$ for the corresponding monotone path (The two step sets describe the same objects: $U \mapsto N$ , $D \mapsto E$ is a bijection matching the diagonal $y = x$ with the level $0$ ).

[L2]

The total number of monotone paths from $(0, 0)$ to $(n, n)$ is $(\binom{2 n}{n})$ ( $∣ M ((0, 0), (m, n)) ∣ = (\binom{m + n}{n})$ ).

[L3]

$(n + 1) C_{n} = (\binom{2 n}{n})$ ( $(n + 1) C_{n} = (\binom{2 n}{n})$ ).

Refutation

technique · direct

1.1

At $n = 2$ there are $(\binom{4}{2}) = 6$ monotone paths from $(0, 0)$ to $(2, 2)$ by [L2].

1.2

The weakly-below ones are exactly EENN and ENEN, so there are $2$ of them.

given

2.1

Half of the total would be $3$ , not $2$ , so the claim is false already at $n = 2$ . The general reason is that some monotone paths cross the diagonal and therefore belong to neither weak half-plane, so the naive symmetry "below equals above equals half of all paths" breaks down.

step 1.1step 1.2L1L3∎

Remarks

The true count is $C_{n}$ , not $(\binom{2 n}{n}) / 2$ . At $n = 2$ that is $C_{2} = 2$ , exactly as the two listed paths show.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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, §§10.2–10.3 (standard reference, not scraped)

FALSE: the monotone paths from (0,0)(0,0) to (n,n)(n,n) staying weakly below the diagonal are exactly half of all monotone paths