On Unique-Factorization Combinatorics via the Pascal Triangle

by Frank C. Fung ( 1st published in OCtober, 2012. )

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Section XVI - The formal Combinatorics Equation for { Category [ T ] }

Summary for the Section :

• In this Section , we shall list-out the formal Combinatorics Equation
  • for the Full Count of { Category [ T ] Composite Numbers } that can be formulated
    • with a given set of [ H ] distinct-and-diffrent prime-numbers .

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The Official Formula :

• For the above-said Combinatorics Equation ,
  • there are in fact two (2) cases we need to look at , namely :
    • Case 1 : when [ H ] is less-than-or-equal to [ T ] ;

    • Case 2 : when [ H ] is greater than [ T ] .

• For our Combinatorics Equations to follow , let us first define :
  • the value [ Z ] as the value for the Full Count ,

    and

  • the values [ A-sub-1 ] thru [ A-sub-T ] , i.e. :
    • 

      as the values on the { Row [ T ] } of the Pascal Triangle .

  Then :

  • For Case 1 , we have :
    • 

      or alternately the same equation in this format :

    • 

  • For Case 2 , we have :
    • 

      or alternately the same equation in this format :

    • 

• This then completes our investigation on the Full Count for { Category [ T ] } ,
  • for the time-being .

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What's Next :

• In the next few Sections , we shall be putting the above Combinatorics Equations into a Matrix Notation format ,
  • in preparation for a comment on Symmetry in Section XXI .

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go to the next section : Section XVII - Creating the 11 x 11 [ PASCAL Matrix ]

go to the last section : Section XV - Summarizing on the Formulation Process

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08