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Permutations and Weak order OPAHC : Partial Order and Hopf Algebras in Combinatorics

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(1)

Permutations and Weak order

OPAHC : Partial Order and Hopf Algebras in Combinatorics

(2)

Permutations: an easy way to understand

Shuffling a deck of cards

(3)

Permutations

Let’s take less cards: 8

(4)

Permutations

Let’s take less cards: 8

12345678 15738642 18472635

(5)

Permutations

Definition: A permutation of size n is a word on the alphabet[n] := {1,...,n} where each letter appears exactly once.

Example, all 6 permutations of size 3 : 123, 132, 213, 231, 312, 321

(6)

Bubble sort

15738642

(7)

Bubble sort

15738642 -> 15378642

(8)

Bubble sort

15738642 -> 15378642 -> 13578642

(9)

Bubble sort

15738642 -> 15378642 -> 13578642 -> 13576842

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Bubble sort

15738642 -> 15378642 -> 13578642 -> 13576842 -> …… -> 12345678

(11)

Bubble sort

15738642 -> 15378642 -> 13578642 -> 13576842 -> …… -> 12345678 15738642 -> 15738624 -> 15738264 -> 15738246 -> …… -> 12345678

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Bubble sort

15738642 -> 15378642 -> 13578642 -> 13576842 -> …… -> 12345678 15738642 -> 15738624 -> 15738264 -> 15738246 -> …… -> 12345678 Less letters: 3

321 -> 231 -> 213 -> 123 321 -> 312 -> 132 -> 123

(13)

Bubble sort

15738642 -> 15378642 -> 13578642 -> 13576842 -> …… -> 12345678 15738642 -> 15738624 -> 15738264 -> 15738246 -> …… -> 12345678 Less letters: 3

321 -> 231 -> 213 -> 123 321 -> 312 -> 132 -> 123

(14)

Weak order size 4

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Weak order size 4 (permutohedron)

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