OpenMath Content Dictionary: combinat1

Canonical URL:

http://www.openmath.org/cd/combinat1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

combinat1.ocd

CD as XML Encoded OpenMath:

combinat1.omcd

Defines:

Bell, binomial, Fibonacci, multinomial, Stirling1, Stirling2

Date:

2004-03-30

Version:

3

Review Date:

2006-03-30

Status:

experimental

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
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          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
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This CD defines some basic combinatorics definitions.

Written by S. Dalmas (INRIA Sophia Antipolis) for the Esprit OpenMath
project. 

binomial

Role:

application

Description:

The binomial coefficients. binomial(n, m) is the number of ways of choosing m objects from a collection of n distinct objects without regard to the order.

Commented Mathematical property (CMP):

binomial(n,m) = n!/(m!*(n-m)!)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="combinat1" name="binomial"/>
      <OMV name="n"/>
      <OMV name="m"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="integer1" name="factorial"/>
        <OMV name="n"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMA>
          <OMS cd="integer1" name="factorial"/>
	  <OMV name="m"/>
        </OMA>
        <OMA>
          <OMS cd="integer1" name="factorial"/>
	  <OMA>
	    <OMS cd="arith1" name="minus"/>
	    <OMV name="n"/>
	    <OMV name="m"/>
	  </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (binomial ( n, m) , divide (factorial ( n) , times (factorial ( m) , factorial (minus ( n, m) ) ) ) )

$$\left(\begin{array}{c}n\\ m\end{array}\right)=\frac{n!}{m!(n-m)!}$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="binomial"/>
    <OMI> 4 </OMI>
    <OMI> 2 </OMI>
  </OMA>
  <OMI> 6 </OMI>
</OMA>
</OMOBJ>

eq (binomial ( 4 , 2 ) , 6 )

$$\left(\begin{array}{c}4\\ 2\end{array}\right)=6$$

Signatures:

sts

multinomial

Role:

application

Description:

The multinomial coefficient, multinomial(n, n1, ... nk) is the number of ways of choosing ni objects of type i (i from 1 to k) without regard to order, in such a way that the total number of objects chosen is n. multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!).

Commented Mathematical property (CMP):

multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!) where n=n1+...+nk

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMS cd="combinat1" name="multinomial"/>
	<OMA>
	  <OMS cd="list2" name="cons"/>
	  <OMV name="n"/>
	  <OMV name="nList"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMA>
	  <OMS cd="integer1" name="factorial"/>
	  <OMV name="n"/>
	</OMA>
        <OMA>
          <OMS cd="fns2" name="apply_to_list"/>
          <OMS cd="arith1" name="times"/>
          <OMV name="nList2"/>
        </OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="linalg1" name="vector_selector"/>
	<OMV name="i"/>
	<OMA>
	  <OMS cd="fns2" name="apply_to_list"/>
	  <OMS cd="linalg2" name="vector"/>
	  <OMV name="nlist2"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="integer1" name="factorial"/>
	<OMA>
	  <OMS cd="linalg1" name="vector_selector"/>
	  <OMV name="i"/>
	  <OMA>
	    <OMS cd="fns2" name="apply_to_list"/>
	    <OMS cd="linalg2" name="vector"/>
	    <OMV name="nList"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
	<OMS cd="arith1" name="plus"/>
	<OMV name="nList"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

and (eq (apply_to_list (multinomial, cons ( n, nList) ) , divide (factorial ( n) , apply_to_list (times, nList2) ) ) , eq (vector_selector ( i, apply_to_list (vector, nlist2) ) , factorial (vector_selector ( i, apply_to_list (vector, nList) ) ) ) , eq ( n, apply_to_list (plus, nList) ) )

$$\mathrm{apply\_to\_list}(\left(\begin{array}{c}\mathrm{cons}(n,\mathrm{nList})\\ \end{array}\right),\mathrm{cons}(n,\mathrm{nList}))=\frac{n!}{\mathrm{apply\_to\_list}(\times ,{\mathrm{nList}}_2)}\wedge {\mathrm{apply\_to\_list}(\mathrm{vector},{\mathrm{nlist}}_2)}_i={\mathrm{apply\_to\_list}(\mathrm{vector},\mathrm{nList})}_i!\wedge n=\mathrm{apply\_to\_list}(+,\mathrm{nList})$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="multinomial"/>
    <OMI> 8 </OMI>
    <OMI> 2 </OMI>
    <OMI> 3 </OMI>
    <OMI> 3 </OMI>
  </OMA>
  <OMI> 560 </OMI>
</OMA>
</OMOBJ>

eq (multinomial ( 8 , 2 , 3 , 3 ) , 560 )

$$\left(\begin{array}{c}8\\ 233\end{array}\right)=560$$

Signatures:

sts

Stirling1

Role:

application

Description:

The Stirling numbers of the first kind. (-1)^(n-m)*Stirling1(n,m) is the number of permutations of n symbols which have exactly m cycles. Note that there are a few slightly different definitions of these numbers.

Commented Mathematical property (CMP):

Stirling1(n,m) = the sum k=0 to n-m of (-1)^k * binomial(n-1+k, n-m+k) * binomial(2n-m,n-m-k) * Stirling2(n,m)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="combinat1" name="Stirling1"/>
      <OMV name="n"/>
      <OMV name="m"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="sum"/>
      <OMA>
        <OMS cd="interval1" name="integer_interval"/>
        <OMS cd="alg1" name="zero"/>
        <OMA>
	  <OMS cd="arith1" name="minus"/>
	  <OMV name="n"/>
	  <OMV name="m"/>
	</OMA>
      </OMA>
      <OMBIND>
	<OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="k"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMA>
	    <OMS cd="arith1" name="power"/>
	    <OMA>
	      <OMS cd="arith1" name="unary_minus"/>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMV name="k"/>
	  </OMA>
	  <OMA>
	    <OMS cd="combinat1" name="binomial"/>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="n"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMV name="k"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="n"/>
		<OMV name="m"/>
	      </OMA>
	      <OMV name="k"/>
	    </OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="combinat1" name="binomial"/>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMA>
		<OMS cd="arith1" name="times"/>
		<OMI> 2 </OMI>
		<OMV name="n"/>
	      </OMA>
	      <OMV name="m"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMA>
		<OMS cd="arith1" name="minus"/>
		<OMV name="n"/>
		<OMV name="m"/>
	      </OMA>
	      <OMV name="k"/>
	    </OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="combinat1" name="Stirling2"/>
	    <OMV name="n"/>
	    <OMV name="m"/>
	  </OMA>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (Stirling1 ( n, m) , sum (integer_interval (zero, minus ( n, m) ) , lambda [ k ] . (times (power (unary_minus (one) , k) , binomial (plus (minus ( n, one) , k) , plus (minus ( n, m) , k) ) , binomial (minus (times ( 2 , n) , m) , minus (minus ( n, m) , k) ) , Stirling2 ( n, m) ) ) ) )

$$\mathrm{Stirling1}(n,m)=\sum _{k=0}^{n-m}{-1}^k\left(\begin{array}{c}n-1+k\\ n-m+k\end{array}\right)\left(\begin{array}{c}2n-m\\ n-m-k\end{array}\right)\mathrm{Stirling2}(n,m)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="Stirling1"/>
    <OMI> 10 </OMI>
    <OMI>  7 </OMI>
  </OMA>
  <OMI> -9450 </OMI>
</OMA>
</OMOBJ>

eq (Stirling1 ( 10 , 7 ) , -9450 )

$$\mathrm{Stirling1}(10,7)=-9450$$

Signatures:

sts

Stirling2

Role:

application

Description:

The Stirling numbers of the second kind. Stirling2(n, m) is the number of partitions of a set with n elements into m non empty subsets. Note that there are a few slightly different definitions of these numbers.

Commented Mathematical property (CMP):

Stirling2(n,m) = 1/m! * the sum from k=0 to m of (-1)^(m-k) * binomial(m,k) * k^n

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="combinat1" name="Stirling2"/>
      <OMV name="n"/>
      <OMV name="m"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="alg1" name="one"/>
	<OMA>
	  <OMS cd="integer1" name="factorial"/>
	  <OMV name="m"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="sum"/>
	<OMA>
	  <OMS cd="interval1" name="integer_interval"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="m"/>
	</OMA>
	<OMBIND>
	  <OMS cd="fns1" name="lambda"/>
	  <OMBVAR>
	    <OMV name="k"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMA>
	        <OMS cd="arith1" name="unary_minus"/>
	        <OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
	        <OMV name="m"/>
	        <OMV name="k"/>
	      </OMA>
	    </OMA>
	    <OMA>
	      <OMS cd="combinat1" name="binomial"/>
	      <OMV name="m"/>
	      <OMV name="k"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMV name="k"/>
	      <OMV name="n"/>
	    </OMA>
	  </OMA>
	</OMBIND>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (Stirling2 ( n, m) , times (divide (one, factorial ( m) ) , sum (integer_interval (zero, m) , lambda [ k ] . (times (power (unary_minus (one) , minus ( m, k) ) , binomial ( m, k) , power ( k, n) ) ) ) ) )

$$\mathrm{Stirling2}(n,m)=\frac{1}{m!}\sum _{k=0}^m{-1}^{(m-k)}\left(\begin{array}{c}m\\ k\end{array}\right)k^n$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="Stirling2"/>
    <OMI> 7 </OMI>
    <OMI> 3 </OMI>
  </OMA>
  <OMI> 301 </OMI>
</OMA>
</OMOBJ>

eq (Stirling2 ( 7 , 3 ) , 301 )

$$\mathrm{Stirling2}(7,3)=301$$

Signatures:

sts

Fibonacci

Role:

application

Description:

The Fibonacci numbers, defined by the linear recurrence: Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1). Note that some authors define Fibonacci(0) = 1.

Commented Mathematical property (CMP):

Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="combinat1" name="Fibonacci"/>
	<OMS cd="alg1" name="zero"/>
      </OMA>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="combinat1" name="Fibonacci"/>
	<OMS cd="alg1" name="one"/>
      </OMA>
      <OMS cd="alg1" name="one"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="combinat1" name="Fibonacci"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="combinat1" name="Fibonacci"/>
	  <OMV name="n"/>
	</OMA>
	<OMA>
	  <OMS cd="combinat1" name="Fibonacci"/>
	  <OMA>
	    <OMS cd="arith1" name="minus"/>
	    <OMV name="n"/>
	    <OMS cd="alg1" name="one"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

and (eq (Fibonacci (zero) , zero) , eq (Fibonacci (one) , one) , eq (Fibonacci (plus ( n, one) ) , plus (Fibonacci ( n) , Fibonacci (minus ( n, one) ) ) ) )

$$\mathrm{Fibonacci}\left(0\right)=0\wedge \mathrm{Fibonacci}\left(1\right)=1\wedge \mathrm{Fibonacci}\left(n+1\right)=\mathrm{Fibonacci}\left(n\right)+\mathrm{Fibonacci}\left(n-1\right)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="Fibonacci"/>
    <OMI> 10 </OMI>
  </OMA>
  <OMI> 55 </OMI>
</OMA>
</OMOBJ>

eq (Fibonacci ( 10 ) , 55 )

$$\mathrm{Fibonacci}\left(10\right)=55$$

Signatures:

sts

Bell

Role:

application

Description:

The Bell numbers: Bell(n) is the total number of possible partitions of a set of n elements.

Commented Mathematical property (CMP):

Bell(n) = the sum from k=0 to n of Stirling2(n,k)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="combinat1" name="Bell"/>
      <OMV name="n"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="sum"/>
      <OMA>
        <OMS cd="interval1" name="integer_interval"/>
        <OMS cd="alg1" name="zero"/>
        <OMV name="n"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
	<OMBVAR>
	  <OMV name="k"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="combinat1" name="Stirling2"/>
	  <OMV name="n"/>
	  <OMV name="k"/>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

eq (Bell ( n) , sum (integer_interval (zero, n) , lambda [ k ] . (Stirling2 ( n, k) ) ) )

$$\mathrm{Bell}\left(n\right)=\sum _{k=0}^n\mathrm{Stirling2}(n,k)$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA> 
    <OMS cd="combinat1" name="Bell"/>
    <OMI> 7 </OMI>
  </OMA>
  <OMI> 877 </OMI>
</OMA>
</OMOBJ>

eq (Bell ( 7 ) , 877 )

$$\mathrm{Bell}\left(7\right)=877$$

Signatures:

sts