Lexicographic unranking constructs the object at a specified position in lexicographic order. At each step it counts valid completions for each possible next symbol, skips blocks preceding the desired rank, and continues within the selected block.

With the convention used here, $E(n,k)$ counts the alternating permutations $a_1\cdots a_{n+1}$ of $\{1,\dots ,n+1\}$ satisfyingThe direction of the first comparison and the shift in the first-entry index are essential when comparing tables. The recurrence relation isHence the first rows in increasing $k$ are $(1)$, $(0,1)$, $(0,1,1)$ and $(0,1,2,2)$. For $n=2$, the two alternating permutations are $213$ and $312$, explaining the two nonzero entries. The row sum counts all alternating permutations on $n+1$ labels; the last entry counts those on $n$ labels. Reversed-row prefix sums compute the Entringer triangle without enumerating the permutations individually.^[ref]

Let a finite group $G$ act on a finite set $X$, with $c_j(g)$ the number of length-$j$ cycles of a permutation induced by $g$. Its cycle index isFor $q$ colours with weight variables $x_1,\dots ,x_q$, the generating function counting group action orbits of colourings by their colour multiplicities isA colouring fixed by $g$ is constant on every cycle of a permutation. A length-$j$ cycle of a permutation therefore contributes ${\sum }_a{x}_a^j$. Multiplication over the cycles of a permutation and the weighted Burnside lemma prove the formula.^[ref]

For four positions under rotations, the cyclic group has cycle index $(z_1^4+z_2^2+2z_4)/4$. With two colours, substitution givesThus there are six necklaces, with two having exactly two positions of each colour: adjacent equal colours or alternating colours. Setting every $x_a=1$ gives the unweighted count; dividing the number of labelled colourings by $|G|$ would fail because some colourings have larger stabilizers.

The subset zeta transform sends a function $f$ on subsets to $F(S)={\sum }_{T\subseteq S}f(T)$. For subsets of an $n$-element set, all values can be computed in $O(n{2}^n)$ additions by processing one element at a time. Reversing containment gives the superset transform.

Pólya enumeration counts objects under a group action, often by applying the Pólya enumeration theorem to colourings.

Unranking constructs the object at a specified rank in a chosen ordering of a finite class. Lexicographic unranking uses lexicographic order.

A polycube is a face-connected union of finitely many unit cubes in the three-dimensional cubic grid. Its exposed square faces determine its surface area.

Combinatorics on words studies finite and infinite sequences of symbols from an alphabet, especially repetition, ordering and occurrence of shorter words. A finite word is an element of the free monoid generated by its alphabet, with concatenation as its operation. For example, abab has two occurrences of ab and is the square of ab; aba is a palindrome. These questions connect algorithms on words with generating functions and symbolic descriptions of dynamical systems.

A cyclically ordered collection of endpoints paired into chords of a circle. The crossing relation defines a circle graph; two chords with distinct endpoints cross exactly when their endpoints alternate.

The RGB color model represents a color through red, green and blue components. In a normalized additive linear-light representation, $(0,0,0)$ is black and equal full-strength channels define the reference white. A component's meaning depends on the chosen primaries and white point. Encoded sRGB components are nonlinear functions of linear light, so averaging their numeric values is not generally the same as averaging light intensities. Eight-bit RGB stores three channel values with 256 levels each, but that bit count alone does not specify a device-independent gamut.^[ref]

For two equal spheres of radius $a$ separated by a surface gap $h\ll a$, a positive Hamaker constant $A_H$ gives the nonretarded Van der Waals force approximation $U=-A_{H}a/(12h)$ and $F=-dU/dh=-A_{H}a/(12h^2)$. The Derjaguin approximation derives the geometry factor from the planar interaction energy per area $W=-A_H/(12\pi h^2)$ and effective radius $a/2$. A sphere and plane instead have effective radius $a$, doubling the force at the same gap. Molecular repulsion, roughness and retardation limit these continuum formulas.

For gently curved interacting surfaces whose range and closest gap are small compared with their radii, integrate the local planar interaction over nearly parallel surface elements. With $W(s)$ the planar interaction free energy per area and $\Pi (s)=-W^{\prime }(s)$, the local gap for two spheres is $s\simeq h+r^2/(2R_{\mathrm{eff}})$, where $R_{\mathrm{eff}}^{-1}=a_1^{-1}+a_2^{-1}$. Thus $F=2\pi {\int }_0^{\infty }r\Pi (s)dr=2\pi R_{\mathrm{eff}}{\int }_h^{\infty }\Pi (s)ds=2\pi R_{\mathrm{eff}}W(h)$ when $W(\infty )=0$. Equal spheres have $R_{\mathrm{eff}}=a/2$; a sphere and plane have $R_{\mathrm{eff}}=a$. Use consistent surface boundary conditions when supplying the planar interaction.

Resistance of a dispersion to aggregation or irreversible separation. Electrostatic and steric interactions and solvent compatibility can all matter.

An ordered arrangement of colloidal particles. Nearly monodisperse hard-sphere-like particles can order through packing entropy at sufficient concentration, while weak reversible attractions or templates can guide ordering. An irreversibly aggregated solid is not necessarily a colloidal crystal.

The distribution of particle dimensions in an ensemble, with number-, volume- or intensity-weighting specified. Broad size distributions can broaden a melting transition, frustrate colloidal crystallization and bias scattering-based size estimates. A mean size alone is then insufficient.

Aggregation of colloidal particles when encounters lead to persistent sticking. Lowering an electrostatic barrier can change a suspension from reaction-limited to approximately diffusion-limited aggregation.

Aggregation caused when an adsorbing polymer chain binds more than one particle. Incomplete surface coverage can favour bridges, whereas a densely covered, well-solvated surface can instead be sterically stabilized.

A dispersion of droplets of one liquid in another immiscible liquid. Its interfacial energy and kinetic persistence depend on droplet size, interfacial tension and stabilizing species.

A force caused by confinement or restructuring of solvent near interfaces. It can depend on molecular layering and specific surface chemistry and need not have the range or sign of an electrostatic double-layer force.

A fluid colloidal system. A solid-in-liquid example is fine silicon dioxide dispersed in water. The term can also include other fluid colloidal systems, so the dispersed and continuous constituents should be specified.