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Signature
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Signature

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Signature[list]

gives the signature of the permutation needed to place the elements of list in canonical order.

Details

  • The signature of the permutation is (-1)n, where n is the number of transpositions of pairs of elements that must be composed to build up the permutation.
  • If any two elements of list are the same, Signature[list] gives 0.
  • Signature can be used on expressions with any head, not only List.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

In[1]:=1
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https://wolfram.com/xid/0bil61gi-n3h
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bil61gi-o1r
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Find even permutations:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-ghh
Out[1]=1

Rank-3 totally antisymmetric (Levi-Civita) tensor:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-dok
Out[1]=1

Contractions of Levi-Civita tensors:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-ipux2g
Out[1]=1

Properties & Relations  (2)Properties of the function, and connections to other functions

Find components of a 3D cross product:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-f9bt6y
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bil61gi-b86q90
Out[2]=2

Compute a determinant:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-bobw1u
Out[1]=1

Compare with built‐in Det:

In[2]:=2
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https://wolfram.com/xid/0bil61gi-j6ko2q
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

The precision of a number influences its ordering:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-c46r8w
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bil61gi-t20s4
Out[2]=2

Signature evaluates even for symbolic arguments:

In[1]:=1
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https://wolfram.com/xid/0bil61gi-ojdl8v
Out[1]=1

Use Unevaluated to insert an unevaluated Signature:

In[2]:=2
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https://wolfram.com/xid/0bil61gi-dwonqb
Out[2]=2

Use Signature directly inside Table:

In[3]:=3
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https://wolfram.com/xid/0bil61gi-lpy80x
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

In[1]:=1
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https://wolfram.com/xid/0bil61gi-quf
Out[1]=1
Wolfram Research (1988), Signature, Wolfram Language function, https://reference.wolfram.com/language/ref/Signature.html.
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Wolfram Research (1988), Signature, Wolfram Language function, https://reference.wolfram.com/language/ref/Signature.html.

Text

Wolfram Research (1988), Signature, Wolfram Language function, https://reference.wolfram.com/language/ref/Signature.html.

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Wolfram Research (1988), Signature, Wolfram Language function, https://reference.wolfram.com/language/ref/Signature.html.

CMS

Wolfram Language. 1988. "Signature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Signature.html.

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Wolfram Language. 1988. "Signature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Signature.html.

APA

Wolfram Language. (1988). Signature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Signature.html

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Wolfram Language. (1988). Signature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Signature.html

BibTeX

@misc{reference.wolfram_2025_signature, author="Wolfram Research", title="{Signature}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Signature.html}", note=[Accessed: 05-June-2025 ]}

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@misc{reference.wolfram_2025_signature, author="Wolfram Research", title="{Signature}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Signature.html}", note=[Accessed: 05-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_signature, organization={Wolfram Research}, title={Signature}, year={1988}, url={https://reference.wolfram.com/language/ref/Signature.html}, note=[Accessed: 05-June-2025 ]}

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@online{reference.wolfram_2025_signature, organization={Wolfram Research}, title={Signature}, year={1988}, url={https://reference.wolfram.com/language/ref/Signature.html}, note=[Accessed: 05-June-2025 ]}