Binomial: Difference between revisions

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{{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] <math>\tbinom nk</math> between the two [[argument|arguments]]. The argument order <source lang=apl inline>k!n</source> is reversed compared to most of traditional mathematical notation's alternative notations, for example <math>C(n,k)</math> and <math>_nC_k</math>, but not others, like <math>C_n^k</math>. Binomial shares the [[glyph]] <source lang=apl inline>!</source> with the monadic arithmetic function [[Factorial]].
{{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] <math>\tbinom nk</math> between the two [[argument|arguments]]. The argument order <syntaxhighlight lang=apl inline>k!n</source> is reversed compared to most of traditional mathematical notation's alternative notations, for example <math>C(n,k)</math> and <math>_nC_k</math>, but not others, like <math>C_n^k</math>. Binomial shares the [[glyph]] <syntaxhighlight lang=apl inline>!</source> with the monadic arithmetic function [[Factorial]].


== Examples ==
== Examples ==


For non-negative integer arguments, the binomial coefficient <source lang=apl inline>k!n</source> is equal to the number of ways to choose k items out of n distinct items. For example, <source lang=apl inline>3!5</source> is 10 because there are 10 ways to pick 3 items out of 5: 123, 124, 125, 134, 135, 145, 234, 235, 245, 345.
For non-negative integer arguments, the binomial coefficient <syntaxhighlight lang=apl inline>k!n</source> is equal to the number of ways to choose k items out of n distinct items. For example, <syntaxhighlight lang=apl inline>3!5</source> is 10 because there are 10 ways to pick 3 items out of 5: 123, 124, 125, 134, 135, 145, 234, 235, 245, 345.


<source lang=apl>
<syntaxhighlight lang=apl>
       0 1 2 3 4 5!5
       0 1 2 3 4 5!5
1 5 10 10 5 1
1 5 10 10 5 1
</source>
</source>


<source lang=apl inline>k!n</source> also corresponds to the k-th value (zero-indexed) on the n-th row (also zero-indexed) of [[wikipedia:Pascal's triangle|Pascal's triangle]].
<syntaxhighlight lang=apl inline>k!n</source> also corresponds to the k-th value (zero-indexed) on the n-th row (also zero-indexed) of [[wikipedia:Pascal's triangle|Pascal's triangle]].


<source lang=apl>
<syntaxhighlight lang=apl>
       ⍉∘.!⍨ 0,⍳5
       ⍉∘.!⍨ 0,⍳5
1 0  0  0 0 0
1 0  0  0 0 0
Line 24: Line 24:
== Properties ==
== Properties ==


The value of <source lang=apl inline>X!Y</source> equals <source lang=apl inline>(!Y)÷(!X)×!Y-X</source>.
The value of <syntaxhighlight lang=apl inline>X!Y</source> equals <syntaxhighlight lang=apl inline>(!Y)÷(!X)×!Y-X</source>.


<source lang=apl>
<syntaxhighlight lang=apl>
       Alt←{(!⍵)÷(!⍺)×!⍵-⍺}
       Alt←{(!⍵)÷(!⍺)×!⍵-⍺}
       0 1 2 3 4 5 Alt 5
       0 1 2 3 4 5 Alt 5
Line 32: Line 32:
</source>
</source>


In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] <math>\Beta(x,y)</math> becomes closely related to the Binomial, giving the identity <math>\Beta(X,Y)</math>{{←→}}<source lang=apl inline>÷Y×(X-1)!X+Y-1</source>.
In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] <math>\Beta(x,y)</math> becomes closely related to the Binomial, giving the identity <math>\Beta(X,Y)</math>{{←→}}<syntaxhighlight lang=apl inline>÷Y×(X-1)!X+Y-1</source>.


<source lang=apl>
<syntaxhighlight lang=apl>
       1 1.2 1.4 1.6 1.8 2!5
       1 1.2 1.4 1.6 1.8 2!5
5 6.105689248 7.219424686 8.281104786 9.227916704 10
5 6.105689248 7.219424686 8.281104786 9.227916704 10

Revision as of 21:47, 10 September 2022

!

Binomial (!) is a dyadic scalar function which gives the binomial coefficient between the two arguments. The argument order <syntaxhighlight lang=apl inline>k!n</source> is reversed compared to most of traditional mathematical notation's alternative notations, for example and , but not others, like . Binomial shares the glyph <syntaxhighlight lang=apl inline>!</source> with the monadic arithmetic function Factorial.

Examples

For non-negative integer arguments, the binomial coefficient <syntaxhighlight lang=apl inline>k!n</source> is equal to the number of ways to choose k items out of n distinct items. For example, <syntaxhighlight lang=apl inline>3!5</source> is 10 because there are 10 ways to pick 3 items out of 5: 123, 124, 125, 134, 135, 145, 234, 235, 245, 345.

<syntaxhighlight lang=apl> 

     0 1 2 3 4 5!5

1 5 10 10 5 1 </source>

<syntaxhighlight lang=apl inline>k!n</source> also corresponds to the k-th value (zero-indexed) on the n-th row (also zero-indexed) of Pascal's triangle.

<syntaxhighlight lang=apl> 

     ⍉∘.!⍨ 0,⍳5

1 0 0 0 0 0 1 1 0 0 0 0 1 2 1 0 0 0 1 3 3 1 0 0 1 4 6 4 1 0 1 5 10 10 5 1

</source>

Works in: Dyalog APL

Properties

The value of <syntaxhighlight lang=apl inline>X!Y</source> equals <syntaxhighlight lang=apl inline>(!Y)÷(!X)×!Y-X</source>.

<syntaxhighlight lang=apl> 

     Alt←{(!⍵)÷(!⍺)×!⍵-⍺}
     0 1 2 3 4 5 Alt 5

1 5 10 10 5 1 </source>

In multiple implementations where Factorial is extended to use the Gamma function , Binomial is defined to use the above equality for non-integers. In that case, the Beta function becomes closely related to the Binomial, giving the identity <syntaxhighlight lang=apl inline>÷Y×(X-1)!X+Y-1</source>.

<syntaxhighlight lang=apl> 

     1 1.2 1.4 1.6 1.8 2!5

5 6.105689248 7.219424686 8.281104786 9.227916704 10 

     2!3j2

1J5

</source>

Works in: Dyalog APL

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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