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, Fibonacci, Stirling1, Stirling2, binomial, multinomial

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2017-12-31

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.

     The copyright holder grants you permission to redistribute this 
     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 
          document is stated prominently in the derived work.  Moreover if
          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 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">binomial</csymbol><ci>n</ci><ci>m</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="integer1">factorial</csymbol><ci>n</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="integer1">factorial</csymbol><ci>m</ci></apply>
    <apply><csymbol cd="integer1">factorial</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><ci>m</ci></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

combinat1.binomial($n, $m) = integer1.factorial($n) / (integer1.factorial($m) * integer1.factorial($n - $m))

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">binomial</csymbol>
   <cn type="integer">4</cn>
   <cn type="integer">2</cn>
  </apply>
  <cn type="integer">6</cn>
 </apply>
</math>

Prefix

eq (binomial ( 4 , 2 ) , 6 )

Popcorn

combinat1.binomial(4, 2) = 6

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: multinomial] [Last: Bell] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol>
    <csymbol cd="combinat1">multinomial</csymbol>
    <apply><csymbol cd="list2">cons</csymbol><ci>n</ci><ci>nList</ci></apply>
   </apply>
   <apply><csymbol cd="arith1">divide</csymbol>
    <apply><csymbol cd="integer1">factorial</csymbol><ci>n</ci></apply>
    <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="arith1">times</csymbol><ci>nList2</ci></apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalg1">vector_selector</csymbol>
    <ci>i</ci>
    <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="linalg2">vector</csymbol><ci>nlist2</ci></apply>
   </apply>
   <apply><csymbol cd="integer1">factorial</csymbol>
    <apply><csymbol cd="linalg1">vector_selector</csymbol>
     <ci>i</ci>
     <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="linalg2">vector</csymbol><ci>nList</ci></apply>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="fns2">apply_to_list</csymbol><csymbol cd="arith1">plus</csymbol><ci>nList</ci></apply>
  </apply>
 </apply>
</math>

Prefix

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) ) )

Popcorn

fns2.apply_to_list(combinat1.multinomial, list2.cons($n, $nList)) = integer1.factorial($n) / fns2.apply_to_list(arith1.times, $nList2) and linalg1.vector_selector($i, fns2.apply_to_list(linalg2.vector, $nlist2)) = integer1.factorial(linalg1.vector_selector($i, fns2.apply_to_list(linalg2.vector, $nList))) and $n = fns2.apply_to_list(arith1.plus, $nList)

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">multinomial</csymbol>
   <cn type="integer">8</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
   <cn type="integer">3</cn>
  </apply>
  <cn type="integer">560</cn>
 </apply>
</math>

Prefix

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

Popcorn

combinat1.multinomial(8, 2, 3, 3) = 560

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Stirling1] [Previous: binomial] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Stirling1</csymbol><ci>n</ci><ci>m</ci></apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <csymbol cd="alg1">zero</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><ci>m</ci></apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>k</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">power</csymbol>
      <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="alg1">one</csymbol></apply>
      <ci>k</ci>
     </apply>
     <apply><csymbol cd="combinat1">binomial</csymbol>
      <apply><csymbol cd="arith1">plus</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><csymbol cd="alg1">one</csymbol></apply>
       <ci>k</ci>
      </apply>
      <apply><csymbol cd="arith1">plus</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><ci>m</ci></apply>
       <ci>k</ci>
      </apply>
     </apply>
     <apply><csymbol cd="combinat1">binomial</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol>
       <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>n</ci></apply>
       <ci>m</ci>
      </apply>
      <apply><csymbol cd="arith1">minus</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><ci>m</ci></apply>
       <ci>k</ci>
      </apply>
     </apply>
     <apply><csymbol cd="combinat1">Stirling2</csymbol><ci>n</ci><ci>m</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

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) ) ) ) )

Popcorn

combinat1.Stirling1($n, $m) = arith1.sum(interval1.integer_interval(alg1.zero, $n - $m), fns1.lambda[$k -> -(alg1.one) ^ $k * combinat1.binomial($n - alg1.one + $k, $n - $m + $k) * combinat1.binomial(2 * $n - $m, $n - $m - $k) * combinat1.Stirling2($n, $m)])

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Stirling1</csymbol>
   <cn type="integer">10</cn>
   <cn type="integer">7</cn>
  </apply>
  <cn type="integer">-9450</cn>
 </apply>
</math>

Prefix

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

Popcorn

combinat1.Stirling1(10, 7) = -9450

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Stirling2] [Previous: multinomial] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Stirling2</csymbol><ci>n</ci><ci>m</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <csymbol cd="alg1">one</csymbol>
    <apply><csymbol cd="integer1">factorial</csymbol><ci>m</ci></apply>
   </apply>
   <apply><csymbol cd="arith1">sum</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol><csymbol cd="alg1">zero</csymbol><ci>m</ci></apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>k</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="alg1">one</csymbol></apply>
       <apply><csymbol cd="arith1">minus</csymbol><ci>m</ci><ci>k</ci></apply>
      </apply>
      <apply><csymbol cd="combinat1">binomial</csymbol><ci>m</ci><ci>k</ci></apply>
      <apply><csymbol cd="arith1">power</csymbol><ci>k</ci><ci>n</ci></apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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) ) ) ) ) )

Popcorn

combinat1.Stirling2($n, $m) = alg1.one / integer1.factorial($m) * arith1.sum(interval1.integer_interval(alg1.zero, $m), fns1.lambda[$k -> -(alg1.one) ^ ($m - $k) * combinat1.binomial($m, $k) * $k ^ $n])

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Stirling2</csymbol>
   <cn type="integer">7</cn>
   <cn type="integer">3</cn>
  </apply>
  <cn type="integer">301</cn>
 </apply>
</math>

Prefix

eq (Stirling2 ( 7 , 3 ) , 301 )

Popcorn

combinat1.Stirling2(7, 3) = 301

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Fibonacci] [Previous: Stirling1] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="combinat1">Fibonacci</csymbol><csymbol cd="alg1">zero</csymbol></apply>
   <csymbol cd="alg1">zero</csymbol>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="combinat1">Fibonacci</csymbol><csymbol cd="alg1">one</csymbol></apply>
   <csymbol cd="alg1">one</csymbol>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="combinat1">Fibonacci</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol><ci>n</ci><csymbol cd="alg1">one</csymbol></apply>
   </apply>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="combinat1">Fibonacci</csymbol><ci>n</ci></apply>
    <apply><csymbol cd="combinat1">Fibonacci</csymbol>
     <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><csymbol cd="alg1">one</csymbol></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

combinat1.Fibonacci(alg1.zero) = alg1.zero and combinat1.Fibonacci(alg1.one) = alg1.one and combinat1.Fibonacci($n + alg1.one) = combinat1.Fibonacci($n) + combinat1.Fibonacci($n - alg1.one)

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Fibonacci</csymbol><cn type="integer">10</cn></apply>
  <cn type="integer">55</cn>
 </apply>
</math>

Prefix

eq (Fibonacci ( 10 ) , 55 )

Popcorn

combinat1.Fibonacci(10) = 55

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Bell] [Previous: Stirling2] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Bell</csymbol><ci>n</ci></apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol><csymbol cd="alg1">zero</csymbol><ci>n</ci></apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>k</ci></bvar>
    <apply><csymbol cd="combinat1">Stirling2</csymbol><ci>n</ci><ci>k</ci></apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

combinat1.Bell($n) = arith1.sum(interval1.integer_interval(alg1.zero, $n), fns1.lambda[$k -> combinat1.Stirling2($n, $k)])

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="combinat1">Bell</csymbol><cn type="integer">7</cn></apply>
  <cn type="integer">877</cn>
 </apply>
</math>

Prefix

eq (Bell ( 7 ) , 877 )

Popcorn

combinat1.Bell(7) = 877

Rendered Presentation MathML

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

Signatures:

sts

---

[First: binomial] [Previous: Fibonacci] [Top]