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# [Solved]: Which permutations can not be obtained by moving elements through two stacks?

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Problem Detail:

I have a Stack1 which has the entries a,b,c ( with a on the Top) and Stack2 which is empty.The condition is.

An entry pooped out of the stack1 can be printed immediatly or pushed to stack2. An Entry popped out of the stack2 can only be printed.

In this Arrangement, which of the permutation of a,b,c is not possible.

I tried diagrammatically by popping out the initial stack and found the first arrangement as c,a,b...Then i mind a formula taught in Permutation's course as (n-1)! possibilites so came up with answer of 5.

so, which arrangement will be not possible.??

#### Answered By : Hendrik Jan

This process is called stack sorting. Wikipedia has a page on the subject.

Here you read that the number of possible sequences when the input is $1,2,\dots,n$ equals the Catalan number $C_n = \frac{1}{n+1} {2n\choose n}$, which differs from your guess.

In fact there is a nice characterization of the orders which are possible, they avoid the permutation pattern $231$. However your problem is "backwards", not the strings that can be sorted using a single stack, but the strings that can be obtained that way.