Length

The number of elements in the collection.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageLength!↗

Length(collection)the number of elements in the collection.

SageMathenumeratio_length(collection)Oscarlength(collection)

Length(expr)the number of top-level operands of any expression, not just a collection.

SageMathenumeratio_length(expr)Oscarlength(expr)

Length(finset)the number of members of a Finset, its cardinality.

SageMathenumeratio_length(finset)Oscarlength(finset)

Typed signatures

Type	Package	Applies
(any) -> infinity | integer	combinatoricsreplaces compute-engine	always
(any) -> infinity | integer	compute-engine	always

Domain: Collections

Details

• Works on any collection head, not just $List$ — e.g. $Set$.
• Additive over concatenation: $Length(Join(A, B)) = Length(A) + Length(B)$. See Join.
• An atom has no parts, so its length is 0 — it isn't a type error.
• Any other expression's length is its number of top-level operands, e.g. the number of terms in a sum.
• See Count to count occurrences of a specific value instead of every element.
• A Finset is read as the subset it is, so its length is its number of members. compute-engine has no Cardinality head, so this is its cardinality.

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Length}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$4$

live ↗

In$\mathrm{Length}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$5$

Works on any collection, not just $List$

live ↗

In$\mathrm{Length}(\lbrace1, 2, 3\rbrace)$Out$3$

live ↗

In$\mathrm{Length}(\bigl\lbrack a, b, c, d\bigr\rbrack)$Out$4$

An empty association has no entries

live ↗

In$\mathrm{Length}(\mathrm{Association}())$Out$0$

Scope #

An atom has no parts, so its length is 0

live ↗

In$\mathrm{Length}(5)$Out$0$

The empty list has length 0

live ↗

In$\mathrm{Length}(\bigl\lbrack\bigr\rbrack)$Out$0$

Of a matrix, the number of rows

live ↗

In$\mathrm{Length}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack, \bigl\lbrack5, 6\bigr\rbrack\bigr\rbrack)$Out$3$

A $Set$ holds each element once, so a repeated element counts once

live ↗

In$\mathrm{Length}(\lbrace1, 1, 2\rbrace)$Out$2$

Any expression: the number of terms in a sum

live ↗

In$\mathrm{Length}(a+b+c+d)$Out$4$

The number of arguments of a call, counting only the top level

live ↗

In$\mathrm{Length}(f(g(x, y), z))$Out$2$

A symbol such as $\pi$ is an atom, of length 0

live ↗

In$\mathrm{Length}(π)$Out$0$

A lazy Range is counted without being materialised

live ↗

In$\mathrm{Length}(1..20)$Out$20$

A symbol is atomic and has no parts

live ↗

In$\mathrm{Length}(x)$Out$0$

Integers are atoms, so no parts

live ↗

In$\mathrm{Length}(123\,456)$Out$0$

A rational literal counts as an atom here

live ↗

In$\mathrm{Length}(\frac{1}{10})$Out$0$

A complex number literal is atomic

live ↗

In$\mathrm{Length}(3+i)$Out$0$

A string is atomic, not a list of characters

live ↗

In$\mathrm{Length}(\text{string})$Out$0$

A Finset has its cardinality: the subset $\{1,3,4\}$ of $\{1,\dots,6\}$ has 3 members

live ↗

In$\mathrm{Length}(\operatorname{Finset}(\operatorname{Tuple}(6,\bigl\lbrack1, 3, 4\bigr\rbrack)))$Out$3$

The empty Finset has no members

live ↗

In$\mathrm{Length}(\operatorname{Finset}((4,\bigl\lbrack\bigr\rbrack)))$Out$0$

Applications #

For a sum, the number of terms

live ↗

In$\mathrm{Length}(x^9+x^2+x+1)$Out$4$

$1000!$ has $2568$ decimal digits

live ↗

In$\mathrm{Length}(\mathrm{IntegerDigits}(1\,000!))$Out$2\,568$

Row lengths of a ragged list

live ↗

In$\mathrm{Map}(\mathrm{Length}, \bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack x\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack2, 3, 1\bigr\rbrack$

live ↗

In$\mathrm{Length}(\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, k, f\bigr\rbrack\bigr\rbrack)$Out$2$

Properties #

$Length(Join(A, B)) = Length(A) + Length(B)$. See Join

live ↗

In$\mathrm{Length}(\mathrm{Join}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5\bigr\rbrack))=\mathrm{Length}(\bigl\lbrack1, 2, 3\bigr\rbrack)+\mathrm{Length}(\bigl\lbrack4, 5\bigr\rbrack)$Out$\mathrm{True}$

Possible issues #

1679 has 1025 squarefree numbers below it, one past compute-engine's own iteration budget (1024) for counting a lazy collection element by element -- this declines rather than silently answer a wrong, short count; see IsSquareFree

live ↗

In$\mathrm{Length}(\mathrm{Filter}(1..1\,679, \mathrm{IsSquareFree}))$Out$\mathrm{Length}(\mathrm{Filter}(1..1\,679, \mathrm{IsSquareFree}))$

An exact fraction is a single atom, so length zero

live ↗

In$\mathrm{Length}(\frac{1}{3})$Out$0$

Complex literals have no parts

live ↗

In$\mathrm{Length}(2+i)$Out$0$

A plain integer has length zero

live ↗

In$\mathrm{Length}(7)$Out$0$

live ↗

In$\mathrm{Length}(\sqrt{7})$Out$0$

About

Examples	31
Checked against	Wolfram Language 29 agree2 differSageMath 21 agree1 differSymPy 18 agree4 differmathlib4 9 agree1 unevaluated8 errorOscar 5 agree
Crosswalk	Wolfram LanguageSageMathOscar

Implementation

mappedmathlib4packages/oracle/src/mappings.ts

The element count of a finite list; an atom or other non-list expression (ours is 0) fails to elaborate.

mappedoscarpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's Length of an atom (not a list) is 0, not an error, matching ours; a bare len() raises on a non-list, so it is guarded.

mappedsympypackages/oracle/src/mappings.ts

Wolfram's Length of an atom (not a list) is 0, not an error, matching ours; a bare len() raises on a non-list, so it is guarded. A sympy expression (not a Python list) has no len but counts its own terms via .args (Add(a,b,c,d).args has 4 elements; a bare Symbol's .args is empty, giving 0 like an atom).

mappedsagepackages/oracle/src/mappings.ts

Wolfram's Length of an atom (not a list) is 0, not an error, matching ours; a bare len() raises on a non-list, so it is guarded. A Sage symbolic expression (not a Python list) has no len but counts its own terms via .operands() — a method, unlike sympy's .args tuple (Add(a,b,c,d).operands() has 4 elements; a bare variable's is empty, giving 0 like an atom).

See also: Count