Generating functions — flashcards

A sign-reversing pairing on partitions that cancels almost every term, leaving the pentagonal number theorem's sparse recurrence. The general principle: a recurrence needing little memory is not an easier problem, it is a harder theorem — the near-total cancellation has to be proved.

Franklin's involution (st/franklin-involution) is a ℤ/2 action on partitions into distinct parts that pairs almost everything freely; its isolated fixed points, at exactly the generalized pentagonal n, are mechanistically identified with the Lefschetz fixed point theorem and Duistermaat–Heckman/Atiyah–Bott localization — a signed sum over a space with a group action reducing to a sum over the fixed locus. This is contrasted, via orbit–stabilizer language, with spontaneous symmetry breaking in physics: a broken vacuum has a reduced stabilizer relative to the full symmetry group, the opposite of Franklin's fixed points, which have full stabilizer (the ℤ/2 action fixes them entirely, rather than moving them to a smaller-symmetry orbit). Goldstone's theorem (every broken continuous symmetry generates a massless mode) is named as physics's standard consequence of the reduced-stabilizer picture, not proved here.

Subbarao's observation that Franklin's involution preserves partition perimeter (λ₁+ℓ(λ)−1) was used to show Franklin's fixed points are exactly the unique perimeter-minimizing partitions into distinct parts — verified for every generalized pentagonal n≤35 — collapsing 'algebraic cancellation' and 'energetic minimization' (st/selection-mechanism-refuted-and-replaced's first and second mechanisms) into one mechanism for this specific case, contradicting the three-way independence claimed there. This collapse was then tested against a second, independently-known Franklin-type involution — Chen and Liu's involution on partitions into distinct parts with odd smallest part, whose fixed points sit at n=k² — by checking five candidate functionals (perimeter, largest part, number of parts, λ₁−ℓ, smallest part) for n≤25; none singles out the squares, with explicit mismatch counts reported for each. The collapse is therefore restricted to the classical Franklin case, not generalized into a new principle: the three-mechanism picture from st/selection-mechanism-refuted-and-replaced is restored, with the explicit caveat that the mechanisms can coincide on particular objects without being identical in general.

The freezing transition (st/freezing-transition-log-correlated-extremes) produces Gumbel-distributed extreme-value statistics; the Gumbel distribution's skewness constant, 12√6·ζ(3)/π³, was derived here from first principles rather than looked up, by identifying the Gumbel distribution's moment generating function as literally Γ(1−t) — so its cumulant generating function is the same log-Gamma Taylor series whose coefficients are zeta values at integers. This ties Apéry's constant ζ(3)'s appearance in the freezing transition's maximum-value statistics directly to the digamma/zeta coefficient identity already established for the completed zeta function's log-derivative (st/xi-log-derivative-local-global-split).

Fabian Franklin, 1881.

The partition recurrence needs only a handful of earlier terms, and the reason is Franklin's involution: almost everything cancels in pairs. Sparse memory is not a lighter problem, it is a heavier theorem — the cancellation has to be proved.

A generating function carries an entire sequence in a single expression; which coefficients you can cheaply extract is a statement about how much of the sequence you needed to store to begin with.

Sequences ↔ elements of a power series ring; convolution of sequences ↔ multiplication. Counting arguments become ring computations.

Encoding a whole sequence as the coefficients of one formal power series, so counting becomes algebra.