On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section XVII - Creating the 11 x 11 [ PASCAL Matrix ]

Summary for the Section :

• In this Section , we shall create the 11 x 11 [ PASCAL Matrix ] .

  And based on the [ PASCAL Matrix ] , we shall create in the next 2 Sections :

  • the 10 x 10 [ ALPHA Matrix ] in Section XVIII ,

    and

  • the 10 x 10 [ ETA Matrix ] in Section XIX .

  Thereafter in Section XX , we shall create the [ Z Matrix ] given by :

  • [ Z Matrix ] = [ ALPHA Matrix ] x [ ETA Matrix ]

  and :

  • the [ Z Matrix ] thus created shall be matrix version of our { Combinatorics Equation } in the last Section ,
    • ripe for further review and analysis in the Sections to follow .

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Creating the [ PASCAL Matrix ] :

• Let us bring-in again the Pascal Triangle ,
  • as per the diagram on-the-left below ;

  and

  • create the corresponding 11 x 11 [ PASCAL Matrix ] ,
    • as per the diagram on-the-right below .

  

  What we have done here is simply to :

  • take each row of Pascal Triangle and put it into the corresponding row in [ PASCAL Matrix ] ,
    • on a row-by-row basis ;

    and

  • for the empty spots to-the-right of the last element [ 1 ] on each row ,
    • we simply fill these up with a bunch of [ 0 ]'s .

• We then also present this set of 2 diagrams below :

  

  • for the exact-same operation as done above ;
    • but possibly for slightly better clarity .

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go to the next section : Section XVIII - Creating the 10 x 10 [ ALPHA Matrix ]

go to the last section : Section XVI - Formalizing the Combinatorics Equation

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08