GATE CS 2024 Set 2 Q.34: Sets, Relations, Functions, Partial Orders and Lattices

Let $P$ be the partial order defined on the set $\{1,2,3,4\}$ as follows

P = \{(x, x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}

The number of total orders on $\{1,2,3,4\}$ that contain $P$ is __________

Your answer

Official answer and solution

Solution

Count the number of topological orderings (linear extensions) of the poset $(\{1, 2, 3, 4\}, P)$ where $1 \prec 2$ , $3 \prec 2$ , and $3 \prec 4$ .

A total order on $\{1, 2, 3, 4\}$ that contains $P$ is a linear extension (topological sort) satisfying the three precedence constraints:

1 \prec 2, \quad 3 \prec 2, \quad 3 \prec 4

Since only $1$ and $3$ have in-degree $0$ (no predecessors), the first element of the total order must be either $1$ or $3$ :

If $1$ is first: the remaining elements $\{2, 3, 4\}$ ${2, 3, 4}$ must satisfy $3 \prec 2$ $3 ≺ 2$ and $3 \prec 4$ $3 ≺ 4$ , so $3$ $3$ must come before both $2$ $2$ and $4$ $4$ . This gives 2 total orders:
- $(1, 3, 2, 4)$
- $(1, 3, 4, 2)$
If $3$ is first: the remaining elements $\{1, 2, 4\}$ ${1, 2, 4}$ only need to satisfy $1 \prec 2$ $1 ≺ 2$ (element $4$ $4$ can be placed in any of the 3 positions among $\{1, 2, 4\}$ ${1, 2, 4}$ ). This gives 3 total orders:
- $(3, 1, 2, 4)$
- $(3, 1, 4, 2)$
- $(3, 4, 1, 2)$

Total number of total orders containing $P$ is $2 + 3 = 5$ .

Final answer

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