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A reading list on Gaussian binomial coefficients, q-binomial coefficients, and their connection to stochastic AUROC

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Part 1

Quick orientation / popular sources

👉 Wikipedia — Gaussian binomial coefficient The best first overview.
👉 MathWorld — q-Binomial Coefficient. A compact formula reference.
👉 NIST DLMF, Chapter 17 — q-Hypergeometric and Related Functions; q-Pochhammer symbols, q-binomial coefficients, and the q-binomial theorem. Not the friendliest introduction, but very reliable for formulas.
👉 An Invitation to Enumeration — q-analogues; One of the clearest online introductions to the topic. This is exactly the language needed for random ROC wandering, where a path is weighted by its area.

Gentle textbook-style entry points
👉 George Andrews, Kimmo Eriksson — Integer Partitions Probably the best textbook-style entry point. It has a chapter on Gaussian polynomials, lattice paths and q-binomial numbers, and the q-binomial theorem. A good source if you want understanding rather than just formulas.
👉 Peter Cameron — Notes on Counting: An Introduction to Enumerative Combinatorics A useful source for q-analogues from the finite-vector-space point of view. It explains why the Gaussian coefficient, when q is a prime power, counts k-dimensional subspaces of an n-dimensional vector space over GF(q). This is not the main route for ROC, but it helps explain why the topic is so large.
👉 Laszlo Babai-style notes — q-combinatorics Short lecture notes.
👉 Herbert Wilf — generatingfunctionology Not specifically about Gaussian binomial coefficients, but extremely useful for learning the general language of generating functions. For our purposes, the key habit is: a discrete distribution can be encoded as the coefficients of a polynomial. That is exactly what happens with the AUROC distribution.
👉 Richard Stanley — Enumerative Combinatorics, Volume 1 The classic serious textbook. Not a light introduction, but an excellent reference if you want to place q-binomial coefficients inside the broader world of inversions, partitions, posets, generating functions, and q-analogues.


Historical sources
👉 H. A. Rothe — Handbuch der reinen Mathematik, 1811 Historically important. The original is not an easy read, but Rothe is often mentioned as one of the early published sources for the q-binomial theorem.
👉 P. A. MacMahon — Combinatory Analysis, 1916 A central source for the combinatorial and generating-function tradition. If your interest is “area under a path,” “inversions,” and “partitions,” MacMahon is closer to our ROC story than the purely analytic q-series tradition.
👉 Leonard Carlitz — A set of polynomials, 1940 Not the easiest entry point, but Carlitz is important in the later q-polynomial and finite-field tradition.
👉 Frank Wilcoxon — Individual Comparisons by Ranking Methods, 1945 This is not about q-binomial coefficients, but it is important for the statistical side of the story. It belongs to the origin of rank-based methods, which are directly connected to AUROC.
👉 Mann, Whitney — On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other, 1947 The key historical source for the Mann–Whitney U statistic. AUROC can be viewed as a normalized Mann–Whitney statistic. In our setting, random tie-breaking inside score blocks gives a finite distribution of a closely related statistic.

Hard but interesting sources

👉 George Andrews — The Theory of Partitions A classic work in partition theory. Useful if you want to understand Gaussian polynomials as generating functions for partitions. Deeper and harder than Andrews–Eriksson.
👉 Gasper, Rahman — Basic Hypergeometric Series also option2 The heavy analytic side of q-series. Not necessary for the ROC project at the beginning, but it is a standard deep reference if you move toward basic hypergeometric series.

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Reading list. Part 2.

👉 Andrews, Askey, Roy — Special Functions A broad and serious reference on special functions. Useful if you want to see the q-binomial theorem as part of the larger world of special functions.
👉 Kathleen O’Hara — Unimodality of Gaussian coefficients: A constructive proof, 1990 A famous paper on the unimodality of Gaussian coefficients. Difficult, but important if you are interested in the shape of the distribution: why the coefficients rise toward the middle and then fall.
👉 Doron Zeilberger — Kathy O’Hara’s Constructive Proof of the Unimodality of the Gaussian Polynomials, 1989 A more explanatory bridge to O’Hara’s proof. Still not easy, but easier than starting with the original paper.
👉 Pak, Panova — Strict unimodality of q-binomial coefficients, 2013 A serious paper on strict unimodality of q-binomial coefficients. Interesting if you want to go beyond “there is a maximum near the centre” and understand finer shape properties of the distribution.
👉 Dhand — A combinatorial proof of strict unimodality for q-binomial coefficients, 2014 A combinatorial proof of strict unimodality in large cases. Hard, but closer in spirit to the combinatorial nature of our problem.

Sources connecting this to ROC AUC

👉 Donald Bamber — The area above the ordinal dominance graph and the area below the receiver operating characteristic graph, 1975 A very important bridge between ROC AUC and the Mann–Whitney U statistic. A good reference for the claim that the area under the ROC curve is a pairwise ranking statistic.
👉 Hanley, McNeil — The meaning and use of the area under a receiver operating characteristic curve, 1982 A classic applied paper on the meaning of ROC AUC. Useful for the interpretation of AUC as the probability that a randomly chosen positive example receives a higher score than a randomly chosen negative example.
👉 Green, Swets — Signal Detection Theory and Psychophysics, 1966 Classic background on signal detection theory and ROC curves. Less about discrete tied score blocks, more about the historical and conceptual origins of ROC analysis.

Suggested reading order for the stochastic ROC project

First:
👉 Wikipedia — Gaussian binomial coefficient
👉 An Invitation to Enumeration — q-analogues

Then:
👉 Andrews, Eriksson — Integer Partitions
👉 Bamber — area below ROC and the Mann–Whitney connection

Then, for a more serious foundation:
👉 Stanley — Enumerative Combinatorics
👉 Mann, Whitney, 1947

And only then, if you want to go deeper into the shape of the distribution:
👉 O’Hara — Unimodality of Gaussian coefficients
👉 Pak, Panova — Strict unimodality of q-binomial coefficients

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