| # P1044 Stack | |
| ## Abstract model | |
| Start with the ordered input sequence $1,2,\ldots,n$, an empty stack, and an | |
| empty output sequence. A legal execution repeatedly pushes the next input item | |
| or pops the stack top. Count the distinct complete output sequences. This count | |
| is the $n$-th Catalan number. The verification case computes the same count by | |
| dynamic programming on the numbers of unprocessed inputs and stacked items; it | |
| restricts $n$ to at most $7$. | |
| ## Problem statement | |
| Stack A has capacity greater than $n$. Two operations are allowed: move the | |
| first remaining operand to the stack top, or move the stack top to the end of | |
| the output sequence. Given the initial operand sequence $1,2,\ldots,n$, output | |
| the number of different output sequences obtainable after all operands have | |
| been processed. | |
| ## Input and output | |
| The input contains one integer $n$. Output the number of possible output | |
| sequences. | |
| ## Constraints | |
| - Original problem: $1 \le n \le 18$. | |
| - Verification entry point: $0 \le n \le 7$. | |
| ## Source | |
| [Luogu P1044, NOIP 2003 Junior, Problem 3](https://www.luogu.com.cn/problem/P1044). | |