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Linear Extensions of Partially Ordered Sets

Let \PXP be a partially ordered set. A linear order L on X is called a linear extension (also, a topological sort) of P, if x\lt y in L whenever x\lt y in P.

Linear Extensions of Partially Ordered Sets

Let \(\PXP\) be a partially ordered set. A linear order \(L\) on \(X\) is called a linear extension (also, a topological sort) of \(P\), if \(x\lt y\) in \(L\) whenever \(x\lt y\) in \(P\). For example, the table displayed in shows that our familiar example \(\bfP_3\) has11 linear extensions.

The classical sorting problem studied in all elementary computer science courses is to determine an unknown linear order \(L\) of a set \(X\) by asking a series of questions of the form: Is \(x\lt y\) in \(L\)? All the well known sorting algorithms (bubble sort, merge sort, quick sort, ) proceed in this manner.

Here is an important special case: determine an unknown linear extension \(L\) of a poset \(\bfP\) by asking a series of questions of the form: Is \(x \lt y\) in \(L\)?

ngoku Akukho calculator icwangcisa le nto, kodwa iinxalenye zayo zibalaseleyo. Zama enye ezantsi, okanye ubhale yakho.

Ukugcina umsebenzi wakho

I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.

Bhalisa Igama elithile

Iimpawu ezisetyenziswa apha

Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.

Imibuzo abantu bebuza

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Iindawo zephepha zitshintshiwe Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Igqityiwe kwaye iphinde yacaciswa apha; iimpazamo zethu.

IiNkqubo Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs