How many sequences possible in stack , if the input(1,2,3,...,n) is in order?

Question

How many permutations can be obtained in the output (in the same order) using a stack assuming that the input is the sequence $(1, 2, 3, 4, 5,\dots, n)$ in that order?

Example If $n=5$, then outputs

$3,4,5,2,1$
$1,2,3,4,5$
$5,4,3,2,1$ etc.

are possible, and outputs

$3,4,5,1,2$
$1,5,2,3,4$
$5,4,3,1,2$ etc.

are impossible.

@hjpotter, because you need to stack $1$ before $2$. And they will go back in the reverse order. — zhoraster, Sep 12 '15 at 09:48
Please edit your post adding that you think these are Catalan numbers (and why do you suspect this). — zhoraster, Sep 12 '15 at 09:49
push(1) , push(2) , push(3) , pop() , push(4) , pop(), push(5) , pop() ,pop() ,*this pop can't give element 1 , infact , it's give element 2*\ so , output 3,4,5,1,2 is not possible . Am I rt ? — hululu, Sep 12 '15 at 09:51

score 1 · Accepted Answer · answered Sep 12 '15 at 10:32

1

Your guess about Catalan numbers is correct. We have a sequence of numbers (pushes), and each of them is popped at some moment. Write a push as a left bracket and a pop as a right bracket, so that, for instance, the sequence $3,4,5,2,1$ corresponds to $((()()()))$. Then we have a one-to-one correspondence with proper bracket sequences, which are enumerated by Catalan numbers.

answered Sep 12 '15 at 10:32

zhoraster

26,096

Thank you very much , I got it . – hululu Sep 12 '15 at 11:00

How many sequences possible in stack , if the input(1,2,3,...,n) is in order?

1 Answers1