Applied Combinatorics Scientist (Enumerative)

About the Role

We build systems where “what works” depends on composition: delay bricks composed into delay polynomials, windows composed into validity decisions, and pipeline stages composed into auditable transformations. The naive space of possibilities explodes fast.

The Applied Combinatorics Scientist makes these spaces countable, searchable, and testable: generate structured families, factor out symmetry/equivalence, design minimal-yet-powerful test corpora, and produce canonical representations that keep the platform from drowning in combinatorial growth.

This is not generic “data science.” It’s discrete math as an enabling engine: reduce exponential blowups, increase coverage, and make compositions controllable.

What You’ll Own

• Enumeration + generation of composition spaces: systematic construction of delay-brick / delay-polynomial families under constraints.
• Equivalence classes + canonical forms: define and implement normal forms under rewrite rules and symmetries.
• Combinatorial coverage for QA: small-but-strong test sets using covering arrays, pairwise/t-wise testing, adversarial constructions.
• Counting + complexity budgeting: quantify growth rates and derive practical bounds to inform engineering/product limits.
• Bridge math to tooling: turn results into generators, canonicalizers, dedupers, and dashboards used by algorithm + validation teams.

What You’ll Do

• Model the objects precisely: delay-brick expressions + rewrite/commutation rules, window-validity condition sets, pipeline graphs.
• Derive enumeration strategies: recurrences, generating functions, constructive grammars, dynamic programming—whatever is practical and sharp.
• Design canonicalization: normal forms and invariants that collapse equivalent constructions so we don’t count the same object 10,000 ways.
• Build usable generators: “give me all unique compositions up to size N under these rules,” with deterministic ordering and reproducible seeds.
• Create minimal test corpora with maximal coverage: structured adversarial cases without brute force.
• Integrate with CI/validation so combinatorially generated cases become regression shields.

Concrete Deliverables

• A combinatorial object spec for a core domain (e.g., delay-polynomial compositions): definitions, equivalences, invariants, constraints.
• A generator + canonicalizer library used by algorithm + test teams.
• A growth/coverage report: explicit counts (or bounds), asymptotics, and practical limits that guide what we attempt at runtime.
• A combinatorial QA suite: covering-array-style test sets + adversarial constructions + minimal counterexample corpora.
• A deduplication/canonicalization pipeline that prevents test/data explosion and enables meaningful benchmarking.

Required Qualifications

• Deep background in enumerative combinatorics (recurrences, generating functions, bijections, constructive counting).
• Strong algorithmic thinking: implement generators, canonicalization strategies, and complexity-aware enumeration.
• Comfort with algebraic/discrete structures: equivalence relations, rewriting systems, symmetry/invariants, graph-like objects.
• Ability to write production-quality research code (tests, determinism, documentation).

Preferred Qualifications

• Experience with rewriting systems / normal forms (term rewriting, e-graphs, canonical labeling).
• Experience with property-based testing and combinatorial testing (pairwise/t-wise coverage, covering arrays).
• Familiarity with graph canonical labeling / isomorphism tooling and methods.
• DSP/estimation intuition helpful for shaping adversarial cases (not required).

How You’ll Be Measured (First 60–90 Days)

• You deliver a working enumerator + deduper for a real composition space we care about.
• You produce a minimal high-coverage QA corpus that catches at least one subtle regression or invalid assumption.
• You quantify a real explosion point and propose a mathematically grounded way to avoid it (canonical forms, pruning, symmetry factoring).
• Your tools get adopted because they’re practical, deterministic, and well-documented.

Working Style

• You treat “let’s brute force it” as a smell and prefer structure-first solutions.
• You like making equivalences explicit so we don’t count the same object 10,000 ways.
• You enjoy turning messy combinations into clean families with sharp boundaries.

Title & Level

Applied Combinatorics Scientist (Enumerative) (senior enabling IC; can scale to Staff/Principal if owning the broader composition + coverage framework), partnering with applied math/DSP, validation, and proof/spec teams.