Log-concavity
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as real rooted polynomial — the same set of essays touches all of them, so they are one junction rather than several.
Coins hidden in the roots
The polynomial that counts permutations by their descents has no product formula, and nothing in the definition of a descent is a coin toss. But every root of the polynomial is real and negative, and a polynomial like that is a product of coins in disguise: each root r is a coin landing heads with chance 1/(1 − r). The descent count of a random permutation is exactly a sum of independent coins nobody can point to — which is why it is bell-shaped, and why its coefficients obey inequalities the inversion count breaks.
The roots every matching polynomial keeps real
Count a graph's matchings by size and make the counts the coefficients of a polynomial. On every one of 35,664 graphs tested, that polynomial has only real roots — a theorem of Heilmann and Lieb from 1972. Count independent sets instead and the roots wander off the axis on a growing share of graphs, but never on a graph without a claw, and matchings are the independent sets of a graph that never has one.
Log-concave with nothing to make it so
The chromatic polynomial of a graph counts its colourings, and its coefficients rise to one peak and fall, each squared at least the product of its neighbours. For a real-rooted polynomial that would be automatic. But by nine vertices fewer than one graph in twenty has only real chromatic roots, and the coefficients stay log-concave anyway — on every graph tested, and, by June Huh's theorem of 2012, on every graph there is.
Named alongside it
The objects these essays reach for when they reach for this one.
Generating functionReal rooted polynomialCentral limit theoremChromatic numberCounterexampleDescentEulerian numberGraph colouringIndependent setInterlacingPerfect matchingPermutation