Saturday, September 5, 2020

Casio fx-9750GIII: Combination Matrices

 Casio fx-9750GIII:  Combination Matrices

Introduction

The program COMBMTRX creates an aligned combination triangle with rows 0 to W-1 and columns 0 to W-1.  The matrix is a square matrix with size W x W.  


1.  Pascal 

n NCR k


2.  Catalan 

(n - k + 1) / (n + 1) * [ n+k NCR n ]


3.  Borel

1 / (n + 1) * [ 2n+2 NCR n-k ] * [ n+k NCR n ]


4.  User Function

f(n, k,  h(n,k) NCR g(n,k))


"Blank" spaces will be filled with 0.  


Casio fx-9750GIII Program:  COMBMTRX

(320 bytes)


Note:  On the program screen, nCr will be symbolized as a bold C


"2020-08-11 EWS"

"COMBINATION MATRIX"

"SIZE"? → W

Identity W → Mat A

Menu "SELECT TYPE","PASCAL",1,"CATALAN",2,"BOREL",3,"USER",4

Lbl 1

"(N)nCr(K)" → fn1

Goto 5

Lbl 2

"(N-K+1)÷(N+1)×(N+K)nCr(N)" → fn1

Goto 5

Lbl 3

"1÷(N+1)×(2N+2)nCr(N-K)×(N+K)nCr(N)" → fn1

Goto 5

Lbl 4

"F(N,K)"? → fn1

Goto 5

Lbl 5

For 1 → I To W

For 1 → J To I

I-1 → N

J-1 → K

fn1 → Mat A[I,J]

Next

Next

Mat A


Examples


Size: 5


Option 1:  Pascal


[[ 1 0 0 0 0 ]

 [ 1 1 0 0 0 ]

 [ 1 2 1 0 0 ]

 [ 1 3 3 1 0 ]

 [ 1 4 6 4 1 ]]


Option 2:  Catalan


[[ 1 0 0 0 0 ]

 [ 1 1 0 0 0 ]

 [ 1 2 2 0 0 ]

 [ 1 3 5 5 0 ]

 [ 1 4 9 14 14 ]]


Option 3:  Borel


[[ 1 0 0 0 0 ]

 [ 2 1 0 0 0 ]

 [ 5 6 2 0 0 ]

 [ 14 28 20 5 0 ]

 [ 42 120 135 70 14 ]]


Option 4:  User

f = (N+K+1)nCr(K) ÷ (N=1)


(in fraction form)


[[ 1 0 0 0 0 ]

 [ 1/2 3/2 0 0 0 ]

 [ 1/3 4/3 10/3 0 0 ]

 [ 1/4 5/4 15/4 35/4 0 ]

 [ 1/5 6/5 21/5 56/5 126/5 ]]


 Source:


Cai, Yue and Yan, Catherine.  "Coutning with Borel's Triangle"  Elsevier B.V. Discrete Mathematics.   November 15, 2018  https://arxiv.org/pdf/1804.01597.pdf

Also:  https://doi.org/10.1016/j.disc.2018.10.031

Retrieved August 9, 2020

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.