SMARANDACHE DUAL SYMMETRIC
FUNCTIONS AND CORRESPONDING NUMBERS OF THE TYPE OF STIRLING NUMBERS OF THE
FIRST KIND
(Amarnath
Murthy, S.E. (E&T),Well Logging Services, Oil and Natural Gas corporation
Ltd.,Sabarmati, Ahmedabad, 380 005 , INDIA.)
In the rising factorial (x+1) (x+2)(x+3). . . (x+n) , the coefficients of different powers of x are the absolute values of the Stirling numbers of the first kind. REF[1].
Let x_{1} , x_{2} , x_{3} , . . . x_{n} be the roots of the equation
(x+1) (x+2)(x+3). . . (x+n) = 0.
Then the elementry symmetric functions are
x_{1} + x_{2} + x_{3} + , . . ., + x_{n }= S x_{1}, ( sum of all the roots )
x_{1}x_{2} + x_{1}x_{3} + . . . x_{n1}x_{n} = S x_{1}x_{2}. ( sum of all the products of the roots taking two at a time )
S x_{1}x_{2}x_{3}…x_{r}
= ( sum of all the products of the roots taking r at a time ) .
In the above we deal with sums of products. Now we define Smarandache Dual symmetric functions as follows.
We take the product of the sums instead of the sum of the products. The duality is evident. As an example we take only 4 variables say x_{1} , x_{2} , x_{3} , x_{4.}_{ }Below is the chart of both types of functions_{ }.
Elementry symmetic funcions (sum of the products) 
Smarandache Dual Symmetric functions (Product of the sums) 
x_{1} + x_{2} + x_{3} + x_{4} 
x_{1}x_{2}x_{3}x_{4} 
x_{1}x_{2 }+ x_{1}x_{3} + x_{1}x_{4} +x_{2}x_{3} + x_{2}x_{4} + x_{3}x_{4} 
(x_{1 }+_{ }x_{2} )_{ }( x_{1}+_{ }x_{3} )( x_{1 }+_{ }x_{4} )(x_{2}+_{ }x_{3} )( x_{2 }+_{ }x_{4} )( x_{3}+_{ }x_{4} ) 
x_{1}x_{2 }x_{3} + x_{1}x_{2}x_{4}+ x_{1}x_{3 }x_{4} + x_{2}x_{3}x_{4} 
(x_{1 }+_{ }x_{2} +x_{3})( x_{1}+_{ }x_{2} + x_{4})( x_{1 }+x_{3} + x_{4} )(x_{2}+_{ }x_{3} +x_{4}) 
x_{1}x_{2}x_{3}x_{4} 
x_{1} + x_{2} + x_{3} + x_{4} 
We define for convenience the product of sums of taking none at a time as 1.
Now if we take x_{r} = r in the above we get the absolute values of the Stirling numbers of the first kind. For the firs column.
24, 50, 35, 10 ,1.
The corresponding numbers for the second column are 10 , 3026, 12600, 24,
1.
The Triangle of the absolute values of Stirling numbers of the first kind
is
1 




1 
1 



2 
3 
1 


6 
11 
6 
1 

24 
50 
35 
10 
1 
The corresponding Smarandache dual symmetric Triangle is
1 




1 
1 



3 
2 
1 


6 
60 
6 
1 

10 
3026 
12600 
24 
1 
The next row (5^{th}) numbers are
15, 240240 , 2874009600, 4233600, 120 , 1.
Following propertiesof the above triangle are visible:
(2) The r^{th }row element of the second leading diagonal contains r! .
(3) The First column entries are the corresponding triangular
numbers.
Readers are invited to find relations between the two triangles.
Application: Smarandache Dual Symmetric functions give us another way of generalising the Arithmetic Mean Geometric Mean Inequality. One can prove easily that
(x_{1}x_{2}x_{3}x_{4})^{1/4} £ [{ (x_{1 }+_{ }x_{2}
)_{ }( x_{1}+_{ }x_{3} )( x_{1 }+_{ }x_{4}
)(x_{2}+_{ }x_{3} )( x_{2 }+_{ }x_{4}
)( x_{3}+_{ }x_{4} )}^{1/6} ] / 2 £
[{(x_{1 }+_{ }x_{2} +x_{3})( x_{1}+_{
}x_{2} + x_{4})( x_{1 }+x_{3} + x_{4}
)(x_{2}+_{ }x_{3} +x_{4}) }^{1/4} ] / 3
£ {x_{1} + x_{2}
+ x_{3} + x_{4} } / 4
The above inequality is generally true can also be established easily.