enumeratio

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Symbol reference ​

Language-reference pages for compute-engine functions, generated from verified example data. Each example is re-evaluated live and pinned by a test, so the docs can't drift from the engine.

Elsewhere ​

Every symbol page opens with where the same thing lives in other systems — Wikipedia, MathWorld, Wikidata, DLMF, Fungrim, the OEIS, FindStat, the Wolfram Language, SageMath, SymPy, mpmath, mathlib4 and more. Nothing there is typed twice: a chip's tooltip says which of the crosswalk's sources it came from.

  • written on the entry, or on one call form when the head means something else at another arity (Zeta(s, a) is Hurwitz's function and links to Hurwitz's pages)
  • curated in the reference package's hand-kept table
  • the enumeratio catalog's own crosswalk, which is where FindStat ids and Sage classes come from; a carrier domain inherits what its collections recorded
  • compute-engine's own Wikidata id — audited, since 41 of the 101 the engine declares are dead or about something else (PlanckConstant pointed at Mount Vesuvius) — and everything that item points at
  • the DLMF's defining equation, found in the handbook's index of notations by the names the head already has
  • FindStat, asked by value: every object of the carrier up to a size, our definition evaluated on each, and the finder's answer kept only where it agrees on everything
  • the OEIS, asked by count: a family's own count kernel for the first sizes, searched, and kept where the terms agree once aligned

Four of those are checks rather than pointers, and the chips say so. A ✓ means the claim was put to a computation — a definition evaluated against theirs, a count against their terms, this head's own examples run in that kernel, or a Fungrim identity instantiated and evaluated on both sides — and the tooltip says how many. A ! means the same computation ran and something disagreed, which is the more interesting outcome: every one of them so far has been a bug on our side, and the entry says which.

The scripts in @enumeratio/reference keep it honest: crosswalk:collect (offline, test-guarded), crosswalk:fetch (the outside indexes), crosswalk:find (FindStat by value, the OEIS by count), crosswalk:verify (Fungrim's identities, numerically), crosswalk:audit (the engine's Wikidata ids) and crosswalk:check (follows every link).

  • Fungrim: the symbol page, and every identity the engine compiled that mentions the head
  • the Wolfram symbol the transpiler vouches for, and the oracle's equivalent call in each kernel — resolved to its documentation anchor through the system's Sphinx inventory

Prose can lean on the same table: <Symbol type="sage">SetComposition</Symbol> is our name, and the link is whatever Sage calls it.

276 heads — 11 defined by a reference expression, 4 primitive on the declared frontier, and 261 not yet placed either way.

Elsewhere: 4031 pointers from 438 of 932 pages into 18 systems — Fungrim entry 2118 ✓698, Wikipedia 276, Wikidata 238, MathWorld 227, Wolfram Language 186 ✓98, OEIS 172 ✓39, Fungrim 141 ✓59, SageMath 126, DLMF 91, FindStat 88 ✓57, Britannica 87, nLab 67, Encyclopedia of Mathematics 66, SymPy 60, mathlib4 44, Rosetta Code 21, mpmath 17, MATLAB 6.

Combinatorics

  • Binomial -- The binomial coefficient $\binom{n}{k}$, the number of k-element subsets of an n-element set.
  • Multinomial -- The multinomial coefficient $\frac{(k_1 + k_2 + \cdots)!}{k_1!\,k_2!\cdots}$ generalises [[Binomial]] to more than two parts.
  • Factorial -- The product of the positive integers up to n, written n!.
  • Factorial2 -- The double factorial $n!!$, the product of every second integer from n down to 1 or 2.
  • CatalanNumber -- The nth Catalan number $C_n$, counting balanced bracket sequences, binary trees, polygon triangulations, and many other structures of size n.
  • Pochhammer -- The rising factorial (a)_n = a(a+1)(a+2)…(a+n−1).
  • Subfactorial -- The number of derangements of n objects: permutations that leave no element in its original position.
  • StirlingS1 -- The signed Stirling number of the first kind, relating falling factorials to powers of n.
  • Stirling -- The Stirling number of the second kind $S(n, k)$: the number of ways to partition an n-set into k non-empty blocks
  • BellNumber -- The nth Bell number, counting the ways to partition a set of n elements.

Sequences

  • Fibonacci -- The nth Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, extended to negative n by the same recurrence.
  • LucasL -- The nth Lucas number: the Fibonacci-style recurrence started from $L_0 = 2$, $L_1 = 1$, closely related to [[Fibonacci]].
  • BernoulliB -- The nth Bernoulli number, a rational sequence appearing in power-sum (Faulhaber) formulas and series expansions.

Number theory

  • GCD -- The greatest common divisor of the arguments.
  • LCM -- The least common multiple of the arguments.
  • Mod -- The remainder of a on division by b.
  • PowerMod -- Modular exponentiation: a^b mod m, computed without forming a^b directly.
  • Totient -- Euler's totient function: the count of integers in 1..n coprime to n.
  • NextPrime -- The next prime strictly greater than n, or the kth prime after n.
  • NthPrime -- The nth prime number (NthPrime(1) = 2).
  • PrimePi -- The prime-counting function: how many primes are ≤ n.
  • IsPrime -- Tests whether n is a prime number.
  • FactorInteger -- The prime factorisation of n as a list of [prime, exponent] pairs.
  • PrimeNu -- The number of distinct prime factors of n.
  • PrimeOmega -- The number of prime factors of n, counted with multiplicity.
  • Divisors -- All positive divisors of n, in increasing order.
  • DivisorSigma -- The sum of the kth powers of the divisors of n.
  • MoebiusMu -- The Möbius function: 0 if n has a squared prime factor, else (-1)^(number of prime factors).
  • IsSquareFree -- Tests whether n has no repeated prime factors.
  • JacobiSymbol -- The Jacobi symbol (n/m), generalising the Legendre symbol to composite m.
  • LegendreSymbol -- The Legendre symbol (n/p): whether n is a quadratic residue mod the odd prime p.
  • KroneckerSymbol -- The Kronecker symbol (a/n): the Jacobi symbol extended to every integer n.
  • ChineseRemainder -- The smallest non-negative integer congruent to each ri modulo the corresponding mi.
  • ExtendedGCD -- The GCD of a and b together with Bézout coefficients x, y such that a·x + b·y = GCD(a, b).
  • MultiplicativeOrder -- The smallest positive k such that a^k ≡ 1 (mod n).
  • IntegerDigits -- The digits of n in the given base (default 10), most significant first.
  • FromDigits -- The integer formed by a list of digits in the given base (default 10).
  • IntegerString -- The string representation of n in the given base (default 10).
  • DigitCount -- Counts of each digit (1 through 9, then 0) occurring in n, base 10 by default.
  • DigitSum -- The sum of the digits of n in the given base (default 10).

Arithmetic

  • Abs -- The absolute value or magnitude of x: $|x|$, its distance from 0.
  • Sign -- The sign of x: -1, 0, or 1 for negative, zero, or positive x.
  • Negate -- The additive inverse of x: $-x$.
  • Square -- The second power of x: $x^2$.
  • Sqrt -- The principal square root of x: $\sqrt{x}$.
  • Root -- The nth root of x: $\sqrt[n]{x} = x^{1/n}$.
  • Floor -- The greatest integer less than or equal to x: $\lfloor x \rfloor$.
  • Ceil -- The least integer greater than or equal to x: $\lceil x \rceil$.
  • Round -- Rounds x to the nearest integer, or to n decimal places.
  • Clamp -- Constrains x to the range [lower, upper].
  • Chop -- Replaces a real number very close to 0 with exact 0.
  • Rationalize -- Finds a simple rational number near x.
  • Max -- The largest of its arguments.
  • Min -- The smallest of its arguments.

Elementary functions

  • Sin -- The sine of x, in radians.
  • Cos -- The cosine of x, in radians.
  • Tan -- The tangent of x, in radians.
  • Cot -- Cotangent, the reciprocal of [[Tan]].
  • Sec -- Secant, the reciprocal of [[Cos]].
  • Csc -- Cosecant, the reciprocal of [[Sin]].
  • Arcsin -- Arcsine, the inverse of [[Sin]] restricted to $[-\pi/2, \pi/2]$.
  • Arccos -- Arccosine, the inverse of [[Cos]] restricted to $[0, \pi]$.
  • Arctan -- Arctangent, the inverse of [[Tan]] restricted to $(-\pi/2, \pi/2)$.
  • Sinh -- Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$.
  • Cosh -- Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
  • Tanh -- Hyperbolic tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$.
  • Exp -- The natural exponential function $e^x$.
  • Ln -- The natural logarithm, the inverse of [[Exp]].
  • Log -- The logarithm of z in base b, defaulting to base 10.
  • Log2 -- The base-2 (binary) logarithm of z.
  • Log10 -- The base-10 (common) logarithm of z.
  • Lb -- The binary logarithm of z, identical to [[Log2]].

Special functions 10 ≝3 primitive

  • Gamma -- The gamma function $\Gamma(z)$, extending the factorial to real and complex arguments; with two arguments, the upper incomplete gamma function.
  • GammaLn -- The natural logarithm of $\Gamma(z)$, avoiding the overflow of computing [[Gamma]] directly for large z.
  • Beta -- Euler's beta function $B(a, b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$, a bridge between [[Gamma]] and [[Binomial]] and the normalizing constant behind the beta distribution.
  • Erf -- The Gauss error function $\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt$, the shape behind the normal distribution's CDF.
  • Erfc -- The complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$, the Gaussian's tail-probability integral.
  • ErfInv -- The inverse error function: $\operatorname{erfinv}(x)$ solves $\operatorname{erf}(y) = x$ for $-1 < x < 1$.
  • Zetanumeric -- The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ for $\operatorname{Re}(s) > 1$, continued analytically elsewhere.
  • HurwitzZeta≝ -- The Hurwitz zeta function $\zeta(s, a) = \sum_{n=0}^{\infty} (n+a)^{-s}$ for $\operatorname{Re}(s) > 1$, continued analytically elsewhere. Provided by `@enumeratio/analytic`.
  • LerchPhi≝ -- The Lerch transcendent $\Phi(z, s, a) = \sum_{n=0}^{\infty} \dfrac{z^n}{(n+a)^s}$ — a common generalization of the polylogarithm and the Hurwitz zeta. Provided by `@enumeratio/analytic`.
  • PolyLog -- The polylogarithm $\operatorname{Li}_s(z) = \sum_{n=1}^{\infty} \dfrac{z^n}{n^s}$, the $a = 1$ slice of the Lerch transcendent.
  • PolyGamma -- The polygamma function $\psi^{(m)}(z) = \dfrac{d^{m+1}}{dz^{m+1}} \ln\Gamma(z)$, the $m$-th derivative of the digamma.
  • Digamma -- The digamma function $\psi(z) = \dfrac{d}{dz}\ln\Gamma(z)$, the logarithmic derivative of [[Gamma]].
  • GammaRegularized -- The regularized upper incomplete gamma function $Q(a, z) = \Gamma(a, z)/\Gamma(a)$, the survival function of the Gamma(a, 1) distribution.
  • BetaRegularized -- The regularized incomplete beta function $I_x(a, b)$, the CDF of the Beta(a, b) distribution and a bridge to the binomial CDF.
  • BarnesG≝ -- The Barnes G-function $G(z)$, the double-gamma function satisfying $G(z+1) = \Gamma(z)\,G(z)$ with $G(1) = 1$ — so $G(n) = \prod_{k=0}^{n-2} k!$, the superfactorial, at positive integers. Provided by `@enumeratio/analytic`.
  • LogBarnesG≝ -- The logarithm of the Barnes G-function, $\ln G(z)$, as an analytic continuation — the form that stays finite where $G$ itself overflows. Provided by `@enumeratio/analytic`.
  • LogGamma≝ -- The log-gamma function $\ln\Gamma(z)$ as an analytic continuation (branch cut on $(-\infty, 0]$), which differs from $\ln(\Gamma(z))$ by multiples of $2\pi i$ off the positive axis. Provided by `@enumeratio/analytic`.
  • ClausenCl≝ -- The Clausen functions $\mathrm{Cl}_n(\theta)$: $\sum_{k\ge1} \sin(k\theta)/k^n$ for even $n$ and $\sum_{k\ge1} \cos(k\theta)/k^n$ for odd $n$ — the imaginary or real part of $\operatorname{Li}_n(e^{i\theta})$. Provided by `@enumeratio/analytic`.
  • DirichletEta≝ -- The Dirichlet eta function $\eta(s) = \sum_{n\ge1} (-1)^{n-1} n^{-s} = (1 - 2^{1-s})\,\zeta(s)$, the alternating zeta — entire, with $\eta(1) = \ln 2$. Provided by `@enumeratio/analytic`.
  • DirichletBeta≝ -- The Dirichlet beta function $\beta(s) = \sum_{n\ge0} (-1)^n (2n+1)^{-s}$, the L-function of the nontrivial character mod 4 — entire, with $\beta(1) = \pi/4$ and $\beta(2) = G$ (Catalan). Provided by `@enumeratio/analytic`.
  • StieltjesGammanumeric -- The Stieltjes constants $\gamma_n$, the coefficients of the Laurent expansion of $\zeta(s)$ at $s = 1$: $\zeta(s) = \dfrac{1}{s-1} + \sum_{n\ge0} \dfrac{(-1)^n}{n!}\gamma_n (s-1)^n$; with a second argument, the generalized $\gamma_n(a)$ for [[HurwitzZeta]]. Provided by `@enumeratio/analytic`.
  • DirichletCharacterkernel -- The $j$-th Dirichlet character modulo $k$ evaluated at $n$: a completely multiplicative, $k$-periodic map $\chi_j: \mathbb{Z} \to \mathbb{C}$, zero where $\gcd(n, k) > 1$. Provided by `@enumeratio/analytic`.
  • DirichletL≝ -- The Dirichlet L-function $L(s, \chi) = \sum_{n\ge1} \chi(n)\,n^{-s}$ of the $j$-th character mod $k$ — the family that contains $\zeta$, $\eta$ and $\beta$, and the setting of Dirichlet's theorem on primes in arithmetic progressions. Provided by `@enumeratio/analytic`.
  • HarmonicNumber≝ -- The harmonic number $H_n = \sum_{k=1}^n \tfrac1k$, exact at a non-negative integer $n$; with a second argument, the generalized $H_n^{(r)} = \sum_{k=1}^n k^{-r}$. Continued off the integers by $H_z = \psi(z+1) + \gamma$. Provided by `@enumeratio/analytic`.

Hypercomplex algebra

  • NonCommutativeMultiply -- The ordered product, for units that anticommute. $\times$ cannot host one: `Multiply` is declared commutative, so canonicalisation sorts its operands before any handler runs and the sign is lost.
  • Norm -- The algebra norm of a hypercomplex element: the determinant of multiplication-by-$z$ on the $2^n$-dimensional space. Reduces to the Gaussian $a^2+b^2$ at one imaginary unit.
  • Basis -- The $2^n$ basis blades of a named algebra, ordered by grade then by generator — so $\mathbb{H}$ comes back as $(1, i, j, k)$.
  • AlgebraSignature -- The signature vector: what each of the algebra's generators squares to, in order — $-1$, $+1$ or $0$.
  • AlgebraDimension -- The dimension $2^n$ of an algebra on $n$ generators — the number of basis blades.
  • Quaternions -- $\mathbb{H}$, the quaternions — which IS $\mathrm{Cl}(0,2)$: two anticommuting generators squaring to $-1$, with $k = f_1f_2$ as the third unit.
  • PowerModList -- Every $x$ in $[0, m)$ with $x^r \equiv a \pmod m$ — and so the place these units already live: the split units of $\mathbb{Z}/m$ are $\mathrm{PowerModList}(1, 1/2, m)$, its imaginary units $\mathrm{PowerModList}(-1, 1/2, m)$.

Diagram algebras

  • Diagram -- A diagram: a set partition of $2n$ points, a top row $1…n$ and a bottom row $1'…n'$, written as blocks of signed labels. The basis element of every algebra below.
  • PartitionAlgebra -- $P_n(\delta)$: every set partition of $2n$ points. The largest of the family — all the others are subalgebras cut out by admitting fewer diagrams. Dimension $B(2n)$, the Bell numbers.
  • BrauerAlgebra -- $B_n(\delta)$: the diagrams that are perfect matchings — every point paired with exactly one other. Dimension $(2n-1)!!$.
  • TemperleyLiebAlgebra -- $TL_n(\delta)$: the PLANAR perfect matchings — the diagrams you can draw in a rectangle without crossings. Dimension $C(n)$, the Catalan numbers.
  • MotzkinAlgebra -- $M_n(\delta)$: the planar diagrams whose blocks have size at most two — so points may also be left unpaired. Dimension $M(2n)$, the Motzkin numbers.
  • OrbitDiagram -- The orbit basis of the partition algebra. A diagram $d_\lambda$ asks for points to be connected; an orbit element $x_\lambda$ asks for them to be connected AND NOTHING ELSE — so the two bases differ by Möbius inversion over the partition lattice.

Numeral systems

  • NumeralSystemShape -- What a system's digits are allowed to look like, in one line.
  • AdicNumeral -- A $b$-adic number: the rational $x$ read in $\mathbb{Z}_b$ (or $\mathbb{Q}_p$ for prime $b$), exact when built from a rational, $+ O(b^{prec})$ when capped. The ring operations work on it.
  • AdicExpansion -- The digits of a $b$-adic number, written with the infinite end on the left: $\ldots 6667$ for $1/3$ in $\mathbb{Z}_{10}$, $0.12$ for $7/25$ in $\mathbb{Q}_5$, $+ O(b^n)$ when capped.
  • AdicValuation -- The $b$-adic valuation $v_b(x)$ — the power of $b$ dividing $x$ — with its companions: the norm $|x|_b = b^{-v}$ and the unit part $u$ in $x = b^v · u$.
  • HenselLift -- The $b$-adic root of a polynomial that reduces to `seed` mod $b$, by Newton's iteration — Hensel's lemma made to run. `AdicSqrt` is the special case $f = x^2 − a$.

Hecke algebras

  • HeckeT -- The basis element $T_w$ of the Iwahori–Hecke algebra $H_n(q)$, indexed by a permutation in one-line notation. Multiplication is the $q$-deformation of the symmetric group's.
  • HeckeSpecialize -- Substitute a value for the deformation parameter. At $q = 1$ the algebra collapses to the group algebra of $S_n$ — which is what makes 'deformation' the right word.

Incidence algebras

  • MoebiusFunction -- The Möbius function $\mu(x,y)$ of a finite poset — the inverse of the zeta function in the incidence algebra. Specialising the poset recovers number theory's $\mu$ and inclusion–exclusion.
  • MoebiusInvert -- Möbius inversion: given $g(y) = \sum_{x \le y} f(x)$, recover $f$. The inverse of [[PosetSumDown]], and the reason the incidence algebra is worth having.
  • PosetElements -- The elements of a finite poset, in a linear extension — the order every other head here indexes values by.

Path algebras

  • QuiverPath -- A directed path in a quiver: where it starts, and the arrow indices it follows. The basis of the path algebra $kQ$, with one trivial path per vertex.
  • QuiverCompose -- Concatenate two paths — or zero, when $q$ does not start where $p$ ends. Most products in a path algebra are zero, and that is the structure rather than a failure.
  • QuiverIsAcyclic -- Whether the quiver has no directed cycle — equivalently, whether its path algebra is finite-dimensional at all.

Hopf algebras

  • QSymM -- The monomial basis $M_\alpha$ of the quasi-symmetric functions, indexed by a composition. The product is the quasi-shuffle, which may ADD two leading parts as well as interleave them.
  • Coproduct -- The coproduct $\Delta$: takes one element to a sum of `HopfTensor` pairs. Deconcatenation for QSym; the multiplicative extension of splitting a part for NSym.
  • Antipode -- The antipode $S$, the last piece of a Hopf algebra: the unique map with $m(S \otimes \mathrm{id})\Delta = \eta\varepsilon$.

Combinatorial Hopf algebras

  • NSymR -- The ribbon basis of NSym, and the fundamental basis `QSymF` of QSym. A composition of $n$ is a subset of $\{1,\ldots,n-1\}$, so the compositions form a Boolean lattice; these bases come from summing over it and inverting.

Group algebras

  • GroupBasis -- A basis element of the group algebra $k[G]$, named by its group element's label. The product is the group's own multiplication, extended bilinearly.
  • ClassSum -- The $k$-th class sum: add up one conjugacy class. Class sums are the basis of the CENTRE of $k[G]$ — a commutative subalgebra of a usually non-commutative algebra.
  • ConjugacyClasses -- The conjugacy classes of a finite group — the orbits of $g \mapsto xgx^{-1}$. Their number is the dimension of the centre of $k[G]$.

The modular group

  • ModularWord -- The unique positive word in $L$ and $R$ of a matrix in $\mathrm{PSL}(2,\mathbb{Z})$ with non-negative entries — which is also its path down the Stern–Brocot tree.
  • ContinuedFraction -- The regular continued fraction $[a_0; a_1, a_2, \dots]$ of $p/q$. Its partial quotients are the run lengths of the rational's Stern–Brocot path, and the $T$-exponents of its matrix — one object under three names.
  • ModularClasses -- Every hyperbolic conjugacy class whose $LR$ word has the given length — equivalently, every closed geodesic of that symbolic period on the modular surface. Conjugation rotates the word, so the classes are binary NECKLACES.
  • RademacherSymbol -- The Rademacher symbol $\Psi$ of a hyperbolic element. By Ghys's theorem it is the linking number of the element's modular knot with the trefoil — and it is also just the number of $R$'s minus the number of $L$'s in its word.
  • FormClassNumber -- How many classes of indefinite binary quadratic forms a discriminant has. A class is not one reduced form but a CYCLE of them, and that cycle is a periodic continued fraction — which is to say, a closed geodesic.

Braids and knots

  • Braid -- A braid in Artin's presentation: $n$ strands and a word whose letter $k$ means $\sigma_k$ and $-k$ means $\sigma_k^{-1}$. Closing it up names a link — and by Alexander's theorem, every link.
  • AlexanderPolynomial -- The Alexander polynomial of a knot, computed from the reduced Burau representation over $\mathbb{Z}[t,t^{-1}]$ and returned as an ordinary expression in $t$. Takes the knot however it is named — as $T(p,q)$, as a braid it closes from, or as a modular word.
  • LorenzBraid -- The braid a closed geodesic of the modular flow draws. By Ghys's theorem the modular knots are exactly the periodic orbits of the Lorenz attractor, and those have a purely combinatorial positive braid.
  • JonesPolynomial -- The Jones polynomial $V(t)$ of a knot. A knot named $T(p,q)$ takes the closed form; anything else goes through the TEMPERLEY–LIEB algebra rather than a matrix representation, where each crossing becomes its two smoothings and each closed loop is worth $\delta = -A^2 - A^{-2}$.

Collections

  • Union -- The set union of the argument collections, de-duplicated.
  • Intersection -- The elements common to all the argument collections.
  • SetMinus -- The elements of set a that are not also in set b.
  • First -- The first element of a collection.
  • Last -- The last element of a collection.
  • At -- The element at the given position; a negative index counts from the end.
  • IndexOf -- The position of the first occurrence of value in the collection.
  • Sort -- The elements of the collection in increasing order.
  • Ordering -- The permutation of indices that would sort the collection into increasing order.
  • Length -- The number of elements in the collection.
  • Count -- The number of elements equal to value in the collection.
  • Join -- The concatenation of the argument collections, in order.
  • Flatten -- The collection with all levels of nested lists merged into one.
  • Append -- The collection with value added as its last element.
  • Partition -- The collection split into consecutive, non-overlapping chunks of length n.
  • Mean -- The arithmetic average of the elements of the collection.
  • Median -- The middle value of the collection once sorted.
  • Mode -- The most frequently occurring element of the collection.
  • Product -- The product of all elements in the collection.
  • Subsets -- The power set of $\{1, …, n\}$ — every subset, as a lazy indexed family of $2^n$.
  • SymmetricGroup -- The $n!$ permutations of $\{1, …, n\}$ as a lazy indexed family, in one-line form.
  • IntegerPartitions -- The partitions of $n$ into positive parts, as a lazy indexed family.
  • DyckPaths -- The Dyck paths of semilength $n$ — balanced up/down words — a lazy family of Catalan many.
  • SetPartitions -- The partitions of the set $\{1, …, n\}$ into non-empty blocks, a lazy indexed family.

Combinatorial collections

  • KSubsets -- The k-element subsets of $\{1, \dots, n\}$ in COLEXICOGRAPHIC order (the combinations) — the order the combinatorial number system ranks them in.
  • Multisets -- The size-k multisets drawn from $\{1, \dots, n\}$ (combinations with repetition).
  • Tuples -- The k-tuples over $\{1, \dots, n\}$: all $n^k$ ordered selections with repetition.
  • Derangements -- The permutations of $\{1, \dots, n\}$ with no fixed point.
  • Involutions -- The self-inverse permutations of $\{1, \dots, n\}$ (only fixed points and 2-cycles).
  • IntegerCompositions -- The compositions of $n$: ordered sequences of positive parts summing to $n$.
  • BinaryTrees -- The binary trees with $n$ internal nodes, as nested lists.

Permutation statistics 1 ≝1 primitive

  • Inversions -- The number of inversions of a permutation $p$: pairs $i < j$ with $p_i > p_j$ (its Kendall-tau distance from the identity).
  • Descents -- The number of descents of $p$: positions $i$ with $p_i > p_{i+1}$.
  • Ascents -- The number of ascents of $p$: positions $i$ with $p_i < p_{i+1}$.
  • MajorIndex≝ -- The major index of $p$: the sum of the descent positions $\sum_{p_i > p_{i+1}} i$.
  • FixedPoints -- The number of fixed points of $p$: positions $i$ with $p_i = i$.
  • CycleCountkernel -- The number of cycles in the disjoint-cycle decomposition of $p$.
  • Excedances -- The number of excedances of $p$: positions $i$ with $p_i > i$.
  • Antiexcedances -- The number of antiexcedances of $p$: positions $i$ with $p_i < i$.
  • Records -- The number of records (left-to-right maxima) of $p$: positions $i$ with $p_i > p_j$ for all $j < i$.
  • Peaks -- The number of peaks of $p$: interior positions $i$ with $p_{i-1} < p_i > p_{i+1}$.
  • Valleys -- The number of valleys of $p$: interior positions $i$ with $p_{i-1} > p_i < p_{i+1}$.
  • MinorIndex -- The minor index of $p$: the sum of the ascent positions $\sum_{p_i < p_{i+1}} i$ (the comajor index).
  • WeakExceedances -- Positions with p(i) >= i.
  • Denert -- Sum of the excedance positions, plus the inversions within each of the excedance and non-excedance subwords.
  • LeftToRightMaxima -- Positions larger than everything before them.
  • LeftToRightMinima -- Positions smaller than everything before them.
  • RightToLeftMaxima -- Positions larger than everything after them.
  • RightToLeftMinima -- Positions smaller than everything after them.
  • FirstDescent -- The smallest descent position, or 0 when p is increasing.
  • LastDescent -- The largest descent position, or 0 when p is increasing.
  • Runs -- Maximal increasing runs — one more than the number of descents.
  • Depth -- Half the total displacement, (1/2) * sum |p(i) - i|.
  • CyclicDescents -- Descents of p read cyclically, counting position n when p(n) > p(1).
  • OccurrencesOf123 -- Triples i < j < k with p(i) < p(j) < p(k).
  • OccurrencesOf132 -- Triples i < j < k with p(i) < p(k) < p(j).
  • OccurrencesOf213 -- Triples i < j < k with p(j) < p(i) < p(k).
  • StackSortable -- 1 when p avoids the pattern 231, 0 otherwise — exactly the stack-sortable permutations.
  • LongestRun -- The length of the longest run of consecutive increases.
  • LargestRunLength -- The length of the longest increasing run (the catalog's second spelling of LongestRun).
  • ReflectionLength -- n minus the number of cycles — the minimum number of transpositions.
  • LargestCycleLength -- The size of the largest cycle.
  • LongestCycleLength -- The size of the largest cycle (the catalog's second spelling).
  • DistinctCycleLengths -- How many distinct cycle sizes occur.
  • TwoCycleCount -- Cycles of size exactly two.
  • ThreeCycleCount -- Cycles of size exactly three.
  • Order -- The order of p in the symmetric group — the lcm of its cycle lengths.
  • LongestIncreasingSubsequence -- The length of a longest increasing subsequence.
  • LongestDecreasingSubsequence -- The length of a longest decreasing subsequence.

Partition statistics

  • LargestPart -- The largest part.
  • MultiplicityOfLargestPart -- How many parts equal the largest.
  • DistinctParts -- How many distinct part sizes occur.
  • EvenParts -- Parts that are even.
  • OddParts -- Parts that are odd.
  • PartsEqualOne -- Parts equal to 1.
  • PartsAtLeastTwo -- Parts of size at least 2.
  • ConjugateOddParts -- Odd parts of the conjugate — equivalently, the distinct part sizes of λ.
  • ConjugateDistinctParts -- Distinct part sizes of the conjugate.
  • DurfeeSquare -- The side of the Durfee square: the largest d with at least d parts of size at least d.
  • ArmOfFirstCell -- The arm of cell (1,1): the first part minus one.
  • LegOfFirstCell -- The leg of cell (1,1): the number of parts minus one.
  • Corners -- Corner cells — parts strictly larger than the next part (the last part always counts).
  • Perimeter -- The perimeter of the Young diagram: largest part plus number of parts.
  • IsSelfConjugate -- 1 when λ equals its conjugate, 0 otherwise.
  • SumOfHookLengths -- The total of all hook lengths.
  • HookProduct -- The product of all hook lengths — the denominator in the hook-length formula.
  • DysonRank -- Largest part minus number of parts.
  • Crank -- The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).

Dyck path statistics

  • Height -- The greatest height the path reaches.
  • DoubleRises -- Occurrences of two consecutive up steps.
  • Returns -- Points where the path comes back to height 0.
  • TouchPointCount -- Points where the path touches the axis — the returns.
  • InteriorReturns -- Returns to height 0 strictly before the end.
  • Hills -- Peaks at height 1 — an up step from the axis immediately followed by a down step.
  • InitialRise -- The length of the opening run of up steps.
  • Area -- The area between the path and the axis: the total of the heights after each step.
  • Coarea -- The complement of the area within the enclosing triangle.
  • LongestAscent -- The longest run of consecutive up steps.
  • LongestDescent -- The longest run of consecutive down steps.
  • Dinv -- The dinv statistic, read from the area sequence.
  • Bounce -- The bounce statistic, which walks the path bouncing off its own peaks.

Set partition statistics

Combinatorial maps

  • Reverse -- The word read backwards.
  • Complement -- Each entry replaced by n + 1 minus itself.
  • Inverse -- The inverse permutation: position of each value.
  • DescentSet -- The positions where the word falls.
  • ToLehmerCode -- Entry i counts the later entries smaller than p(i).
  • CycleType -- The multiset of cycle lengths, as a partition.
  • ReverseComplement -- Reverse, then complement.
  • InverseAfterComplementAfterReverse -- Reverse, then complement, then invert.
  • CyclicShift -- Rotate the word one place to the left.
  • InverseCyclicShift -- Rotate the word one place to the right.
  • PeakSet -- The interior positions that rise then fall.
  • RskInsertion -- The insertion tableau of the RSK correspondence.
  • RskShape -- The common shape of the RSK tableaux: the row lengths, as a partition of n.
  • RskRecording -- The recording tableau of the RSK correspondence, as a row word.
  • Rsk -- The RSK correspondence: the insertion and recording tableaux, as a pair.
  • CyclePartition -- The orbits, as a set partition of the positions.
  • DescentComposition -- The composition of n cut at the descent positions.
  • BinarySearchTree -- The tree built by inserting σ(1), σ(2), ... into an empty binary search tree.
  • KnuthClassRepresentative -- The row reading word of σ's RSK insertion tableau — the canonical word of its Knuth (plactic) class.
  • KrewerasComplement -- w⁻¹c, for w below the long cycle c = (1 2 ... n) in absolute order.
  • ConjugateAfterCycleType -- The conjugate partition of the cycle type.
  • ConjugacyClassRepresentative -- The canonical permutation with the same cycle type.
  • Foata -- Foata's fundamental bijection: cycles rotated to their max, ordered by increasing max, parentheses erased.
  • ArcRepresentation -- Each position linked to the next in its block, or to itself when last.
  • PermutahedronVertex -- A geometric embedding rather than a combinatorial rewrite. Not yet defined.
  • FromPermutation -- A catalog alias whose source collection is ambiguous without the enumeratio map row it came from. Not yet defined.

Carrier domains

85 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.

Compute engine

571 generated pages -- each with the symbol's own description and its crosswalk, and no examples yet.