| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies discrete structures using algebraic methods and studies algebraic structures using combinatorial methods. Includes symmetric functions, Young tableaux, posets, Coxeter groups, association schemes, polynomial invariants, representation theory of symmetric and related groups, generating functions, and algebraic graph theory. Excludes continuous analysis unless discretized or encoded combinatorially; excludes combinatorics without meaningful algebraic structure. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates across discrete, algebraic, and representation-theoretic scales: finite sets, partitions, tableaux, graphs, posets, polynomials, symmetric-group actions, weight spaces, and generating functions encoding infinite combinatorial families. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Partitions; Young diagrams; tableaux; permutations; posets; graphs; monomials; symmetric functions; polynomials; representations; characters; weight vectors; Coxeter generators; adjacency matrices; incidence matrices. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Ordering relations; symmetry; multiplicities; enumeration parameters; dimension counts; eigenvalues of combinatorial matrices; weight and root structures; combinatorial statistics (inversions, descents, major index); algebraic invariants (Schur positivity, unimodality). |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Symmetric functions; representation-graded combinatorial structures; posets and lattices; association schemes; algebraic graph classes; Coxeter groups; matroids; hyperplane arrangements; combinatorial Hopf algebras; Schubert calculus; cluster combinatorics. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Partition shape; tableau filling; permutation pattern; rank or dimension of representation; generating-function parameters; adjacency eigenvalues; weight coordinates; basis choice (Schur, monomial, power-sum, etc.); cell/vertex labels. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded via partitions, tableaux, incidence matrices, adjacency matrices, symmetric-function bases, group actions, polynomial encodings, generating functions, weight diagrams, Coxeter presentations, root systems. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Restricting to finite combinatorial objects; assuming Schur positivity; ignoring torsion phenomena in representation-theoretic settings; using simplified generating functions; focusing on well-behaved graphs or posets; treating Coxeter systems as crystallographic. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Fail in infinite-dimensional representation settings; breakdown in non-crystallographic or wild-type Coxeter groups; failure of positivity assumptions; pathological graph spectra; non-Schur-positive symmetric functions; growth beyond computational tractability. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Discrete structures obey algebraic rules; group actions encode symmetry; generating functions capture combinatorial families; representation dimensions correspond to combinatorial counts; eigenstructures encode combinatorial properties; symmetry bases are complete. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes combinatorial encodings faithfully match algebraic invariants; assumes basis expansions (e.g., Schur functions) are meaningful for structural analysis; assumes representations decompose in combinatorially interpretable ways; assumes manageable computational behavior for discrete families. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Algebraic identities must match combinatorial interpretations; group actions must preserve underlying structures; symmetric-function operations must align with tableau-based rules; poset operations must respect order axioms; polynomial invariants must be basis-consistent. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony between representation theory, symmetric functions, posets, Coxeter theory, generating functions, algebraic graph theory, and Hopf-algebraic structures; compatibility between combinatorial models and algebraic invariants. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Enumeration sequences; symmetric-function expansions; tableau growth and behavior; eigenvalues/eigenvectors of combinatorial matrices; spectra of graphs in association schemes; permutation statistics (inversions, descents, major index); character values in symmetric-group representations; generating-function coefficients; Coxeter group actions. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Difficulty observing behavior of large combinatorial families; computational hardness of symmetric-function expansion; limits on computing Kazhdan–Lusztig polynomials; intractability of large tableau enumeration; spectral limits for very large graphs; inability to fully visualize high-rank Coxeter structures. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Partition sizes; tableau shape parameters; polynomial degrees; eigenvalue magnitudes; multiplicities; generating-function coefficients; rank/dimension; path lengths in posets; Coxeter lengths. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Symmetric-function computation tools; character calculators; tableau generators; graph-spectral solvers; Coxeter-group software; Kazhdan–Lusztig polynomial engines; algebraic combinatorics packages (SageMath, GAP); generating-function analyzers. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Tableau defined by filling rules; symmetric functions defined by basis transformations; character values defined via trace of representation matrices; eigenvalues defined by adjacency/incidence matrices; generating functions defined via combinatorial enumeration; poset relations defined by partial-order axioms. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Constructing tableaux; computing symmetric-function expansions (Schur, monomial, power-sum bases); evaluating generating functions; computing graph spectra; computing character tables; executing permutation statistics; constructing and reducing Coxeter words; generating poset order ideals. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Canonical tableau-generation procedures; standardized symmetric-function basis expansions; uniform sampling of partitions; structured enumeration of posets and graph families; controlled generation of permutations; consistent use of generating-function pipelines; fixed Coxeter rewriting rules. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling partitions of fixed size; random tableau fillings; sampling random graphs from structured families; sampling permutations with given statistics; sampling weight vectors; sampling tableaux and polynomials in Schubert or cluster combinatorics; sampling association-scheme parameters. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Partitions; tableaux; symmetric-function coefficient lists; generating-function expansions; adjacency/incidence matrices; character tables; poset diagrams; polynomial invariant lists; Coxeter words; eigenvalue/eigenvector tables. |
| | Resolution | The granularity or precision with which data is captured. | Determined by tableau size limits; coefficient precision; combinatorial object count; matrix sizes in spectral analysis; maximal depth of generating functions; accuracy of character computation; length/complexity of Coxeter expressions. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Cross-verifying symmetric-function expansions under multiple bases; validating tableau rules; checking consistency of eigenvalues across solvers; verifying generating-function coefficients by independent enumeration; checking representation-theoretic data against known identities; cross-checking Coxeter reduction correctness. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Miscomputed expansions; incorrect tableau generation; spectral approximation errors; faulty generating-function recursion; incorrect character values; misclassified poset relations; errors in Coxeter word reduction; truncation or overflow in large enumerations. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Symmetric-function identities; Schur-positivity relations; RSK insertion rules; hook-length formula behavior; representation–enumeration correspondences; Coxeter relations; spectral regularities in association schemes; unimodality/log-concavity in generating polynomials. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Partition shapes; tableau statistics; character values; eigenvalues of combinatorial matrices; coefficients of symmetric/polynomial invariants (e.g., Tutte, Kazhdan–Lusztig); Möbius invariants of posets; permutation statistics (maj, inv, des). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Group actions generating symmetric structures; insertion algorithms producing tableaux; algebraic operators acting on symmetric functions; adjacency operators driving spectra; poset order governing Möbius inversion; Coxeter generator relations defining reduced words; generating functions encoding recurrences. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Partition → tableau → symmetric-function expansion → representation data; permutation → statistics → generating function → polynomial invariant; graph → adjacency matrix → eigenstructure → combinatorial conclusions; Coxeter group → reduced word → representation or polynomial; poset → Möbius function → combinatorial interpretation. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Partition, tableau, symmetric function, Schur basis, generating function, character, descent statistics, Möbius function, Coxeter system, reduced word, Kazhdan–Lusztig polynomial, association scheme, Hopf algebra, root system. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Symmetric-function bases; representation classes indexed by partitions; poset families (Boolean, Eulerian, graded, distributive); graph families with algebraic structure; Coxeter types (A, B, D, affine); matroids; Schubert/Stanley/cluster combinatorial objects. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Cauchy identity; RSK insertion/deletion equations; hook-length formula; generating-function recurrences; adjacency eigenvalue equations; Möbius inversion formula; Coxeter relations (s_i^2=e), ((s_i s_j)^{m_{ij}} = e); Kazhdan–Lusztig recursion. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Young diagrams/tableaux; symmetric-function lattices; character tables; poset diagrams; graph adjacency/association-scheme matrices; Coxeter complexes; Bruhat order; root systems; combinatorial Hopf-algebra diagrams. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Finite tableaux; low-rank symmetric-function expansions; restricting to type-A Coxeter groups; small-order graphs; posets of limited rank; ignoring torsion/irregular representation behavior; truncated generating functions. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Failures in large ranks; breakdown in wild Coxeter types; explosion of tableau counts; intractable symmetric-function expansions; infinite association schemes; breakdown of unimodality/log-concavity; combinatorial Hopf algebras with non-finite bases. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Representation theory + combinatorics via symmetric functions; Hopf-algebra frameworks unifying combinatorial families; geometric representation theory via Schubert calculus; Coxeter theory unifying groups, polynomials, and posets; analytic combinatorics unifying recurrence structures with algebra. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to geometry (flag varieties, Schubert varieties); representation theory (symmetric groups, Hecke algebras); probability (random tableaux, random partitions); optimization (polytopes, matroids); coding theory (association schemes); statistical physics (exactly solvable models, symmetric polynomials). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Modifying partition shapes; altering tableau rules; adjusting graph parameters; changing symmetric-function bases; varying Coxeter generators; manipulating generating-function variables; modifying representation parameters tied to combinatorial objects. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural tableau growth; tracking spectral behavior of graphs; watching symmetric-function expansion patterns; monitoring generating-function coefficient growth; observing Coxeter reductions; tracking statistics across families of combinatorial structures. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing Schur positivity; verifying symmetric-function identities; checking tableau algorithms (RSK, jeu de taquin); validating unimodality/log-concavity conjectures; testing recurrence relations; validating Coxeter relations and reduced-word behavior. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing symmetric-function expansions using different bases; regenerating tableaux with new RNG seeds; repeating spectral computations; recomputing generating functions with independent algorithms; cross-checking Kazhdan–Lusztig polynomials across tools. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distributions of tableau statistics; studying eigenvalue distributions; comparing generating-function growth rates; estimating asymptotics of partition counts; analyzing permutation statistics; evaluating stability of spectra across graph families. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing symmetric-function bases; contrasting tableau-growth models; comparing spectral properties of graph families; comparing Coxeter systems; evaluating recurrence models; contrasting combinatorial interpretations of representation-theoretic invariants. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting incorrect symmetric-function expansions; tableau-generation errors; miscomputed graph spectra; faulty recurrence outputs; incorrect permutation statistics; wrong Coxeter reductions; overflow/truncation in large enumerations. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding bias toward “nice” partitions/tableaux; ensuring randomness in permutation/graph sampling; avoiding overemphasis on low-rank Coxeter types; preventing basis-dependent conclusions; balancing small/large combinatorial families. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing combinatorial proofs; auditing symmetric-function computations; validating tableau algorithms; cross-checking spectral analyses; reviewing recurrence derivations; confirming representation–combinatorics correspondences. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating combinatorial rules; refining generating-function models; revising symmetric-function identities; modifying tableau interpretations of representations; adjusting Coxeter relations; refining positivity/unimodality assumptions. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Disclosing generating-function conventions, symmetric-function bases, sampling rules, tableau algorithms, Coxeter generators, and computation assumptions; stating all limitations explicitly. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of counterexamples; acknowledging computational limits; ensuring reproducibility; avoiding overclaims of regularity; maintaining rigor in algebraic–combinatorial interpretations. |