On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section XXI - Symnmetry Considerations for the [ Z Matrix ]

Summary for the Section :

• In this Section , we shall be looking at certian characteristics of the [ Z Matrix ] ,
  • including quasi-symmetry to a certain extent .

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Quasi-Symmetry of the [ Z Matrix ] :

• Let us bring-in again the set of 2 diagrams for the [ Z Matrix ] from the last Section :

  

  Let us now draw in a [ Line of Quasi-Symmetry ] ,

  • as marked-off in [ blue color ] in the diagram on-the-left below .

  

  We then bring-in the this version of the Pascal Triangle for a quick comparison :

  

  • It would seem then that the [ Z Matrix ] does exhibit quasi-symmetry to a certain extent .

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The Truncated Matrix :

• Let us now bring-in this set of 2 diagrams , with :
  • the left-hand-side diagram showing the Pascal Triangle , and

  • the right-hand-side diagram showing a 11 x 11 [ Matrix ] based on the Pascal Traingle .

• Let us now bring-in the [ Z Matrix ] for a quick comparison :

• And we can see from these 2 diagrams that :
  • The [ Z Matrix ] is in fact the truncated version of the [ Matrix on-the-right ] ,
    • with the [ 1st row ] and the [ last column ] of the said [ Matrix on-the-right ] being omitted

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go to the next section : Section XXII - Expressing [ z ] as a function of [ H ] and [ T ]

go to the last section : Section XX - Creating the 10 x 10 [ Z Matrix ]

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08