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 x₁ , x₂ , x₃ , . . . x_n be the roots of the equation

(x+1) (x+2)(x+3). . . (x+n) = 0.

Then the elementry symmetric functions are

x₁ + x₂ + x₃ + , . . ., + x_n = S x₁, ( sum of all the roots )

x₁x₂ + x₁x₃ + . . . x_n-1x_n = S x₁x₂. ( sum of all the products of the roots taking two at a time )

S x₁x₂x₃…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 x₁ , x₂ , x₃ , 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)
x1 + x2 + x3 + x4	x1x2x3x4
x1x2 + x1x3 + x1x4 +x2x3 + x2x4 + x3x4	(x1 + x2 ) ( x1+ x3 )( x1 + x4 )(x2+ x3 )( x2 + x4 )( x3+ x4 )
x1x2 x3 + x1x2x4+ x1x3 x4 + x2x3x4	(x1 + x2 +x3)( x1+ x2 + x4)( x1 +x3 + x4 )(x2+ x3 +x4)
x1x2x3x4	x1 + x2 + x3 + x4

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:

1. The leading diagonal contains unity.

(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

(x₁x₂x₃x₄)^1/4 £ [{ (x₁ + x₂ ) ( x₁+ x₃ )( x₁ + x₄ )(x₂+ x₃ )( x₂ + x₄ )( x₃+ x₄ )}^1/6 ] / 2 £

[{(x₁ + x₂ +x₃)( x₁+ x₂ + x₄)( x₁ +x₃ + x₄ )(x₂+ x₃ +x₄) }^1/4 ] / 3 £ {x₁ + x₂ + x₃ + x₄ } / 4

The above inequality is generally true can also be established easily.