Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already cover the safety profile (readOnly, idempotent, non-destructive, openWorld), and the description adds genuinely non-structural information: the call is paid ($0.001 USDC per successful call via x402 on Solana) and the same operation is reachable over HTTPS at a given endpoint. The unclear part is that it names the output fields without explaining their relationship precisely.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is short and the math definition is front-loaded, but the second sentence is repetitive and awkward ('product is that binomial product and value is the Narayana number'), spending words to re-explain output fields. The pricing/endpoint tail is useful and earns its place.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

For a 2-parameter, annotation-covered, output-schema-less tool this is roughly adequate: domain limit, cost, and an alternate transport are all present. What is missing is precise output semantics (whether both 'product' and 'value' are returned, and their types) and any k-bound guidance, which an agent needs to interpret results.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 0%, so the description carries the burden. It states the domain for n (1 to 60) and the formula C(n,k)*C(n,k-1) implies k's role, but k's own range and validity conditions are never stated even though both parameters have min 1 / max 60 in the schema. Partial compensation only.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

The description names the exact mathematical object (the Narayana product C(n,k)*C(n,k-1)) and contrasts it with the plain Narayana number N(n,k), which is the key distinction from the sibling 'narayana'. The wording is slightly garbled ('product is that binomial product and value is the Narayana number'), but an agent can still tell what is computed. It does not explicitly route to narayana / narayana_row / narayana_cows as alternatives.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

There is no statement of when to use this rather than the many sibling Narayana tools (narayana, narayana_row, narayana_cows). The implicit distinction ('n times N(n,k)') is buried in a formula rather than framed as a selection rule. No prerequisites or exclusions are given beyond the n range.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.