permutations card shuffling and representation theory
play

Permutations, Card Shuffling, and Representation Theory Franco - PowerPoint PPT Presentation

Dec 28, 2022 •228 likes •1.71k views

Permutations Card Shuffling Representation theory Permutations, Card Shuffling, and Representation Theory Franco Saliola, Universit du Qubec Montral Based on joint work with: Victor Reiner, University of Minnesota Volkmar Welker,

  1. Permutations Card Shuffling Representation theory � � inc 2 ( τ − 1 σ ) Inc 4 , 2 =   6 5 5 4 4 3 5 4 4 3 3 2 4 3 3 2 2 1 3 2 2 1 1 0 5 6 4 3 5 4 4 5 3 2 4 3 3 2 2 1 1 0 4 3 3 2 2 1     5 4 6 5 3 4 4 3 3 2 2 1 5 4 4 3 3 2 2 3 1 0 2 1     4 3 5 6 4 5 3 2 2 1 1 0 4 5 3 2 4 3 3 4 2 1 3 2     4 5 3 4 6 5 3 4 2 1 3 2 2 3 1 0 2 1 5 4 4 3 3 2     3 4 4 5 5 6 2 3 1 0 2 1 3 4 2 1 3 2 4 5 3 2 4 3      5 4 4 3 3 2 6 5 5 4 4 3 3 2 4 3 1 2 2 1 3 2 0 1     4 5 3 2 4 3 5 6 4 3 5 4 2 1 3 2 0 1 3 2 4 3 1 2     4 3 3 2 2 1 5 4 6 5 3 4 4 3 5 4 2 3 1 0 2 3 1 2     3 2 2 1 1 0 4 3 5 6 4 5 3 2 4 5 3 4 2 1 3 4 2 3     3 4 2 1 3 2 4 5 3 4 6 5 1 0 2 3 1 2 4 3 5 4 2 3     2 3 1 0 2 1 3 4 4 5 5 6 2 1 3 4 2 3 3 2 4 5 3 4      4 3 5 4 2 3 3 2 4 3 1 2 6 5 5 4 4 3 1 2 0 1 3 2     3 2 4 5 3 4 2 1 3 2 0 1 5 6 4 3 5 4 2 3 1 2 4 3     3 2 4 3 1 2 4 3 5 4 2 3 5 4 6 5 3 4 0 1 1 2 2 3     2 1 3 2 0 1 3 2 4 5 3 4 4 3 5 6 4 5 1 2 2 3 3 4     2 1 3 4 2 3 1 0 2 3 1 2 4 5 3 4 6 5 3 4 2 3 5 4     1 0 2 3 1 2 2 1 3 4 2 3 3 4 4 5 5 6 2 3 3 4 4 5     3 4 2 3 5 4 2 3 1 2 4 3 1 2 0 1 3 2 6 5 5 4 4 3     2 3 3 4 4 5 1 2 0 1 3 2 2 3 1 2 4 3 5 6 4 3 5 4      2 3 1 2 4 3 3 4 2 3 5 4 0 1 1 2 2 3 5 4 6 5 3 4     1 2 0 1 3 2 2 3 3 4 4 5 1 2 2 3 3 4 4 3 5 6 4 5     1 2 2 3 3 4 0 1 1 2 2 3 3 4 2 3 5 4 4 5 3 4 6 5    0 1 1 2 2 3 1 2 2 3 3 4 2 3 3 4 4 5 3 4 4 5 5 6

  2. Permutations Card Shuffling Representation theory � � inc 3 ( τ − 1 σ ) Inc 4 , 3 =   4 2 2 1 1 0 2 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 4 1 0 2 1 0 2 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0     2 1 4 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 0     1 0 2 4 1 2 1 0 0 0 0 0 0 2 0 0 1 1 0 1 0 0 0 0     1 2 0 1 4 2 0 1 0 0 0 0 0 1 0 0 0 0 2 0 1 1 0 0     0 1 1 2 2 4 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 1      2 0 1 1 0 0 4 2 2 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0     0 2 0 0 1 1 2 4 1 0 2 1 0 0 1 0 0 0 0 0 1 0 0 0     1 0 0 0 0 0 2 1 4 2 0 1 1 1 2 0 0 0 0 0 0 1 0 0     1 0 0 0 0 0 1 0 2 4 1 2 0 0 0 2 1 1 0 0 0 1 0 0     0 1 0 0 0 0 1 2 0 1 4 2 0 0 0 1 0 0 1 1 2 0 0 0     0 1 0 0 0 0 0 1 1 2 2 4 0 0 0 1 0 0 0 0 0 2 1 1      1 1 2 0 0 0 0 0 1 0 0 0 4 2 2 1 1 0 0 0 0 0 1 0     0 0 0 2 1 1 0 0 1 0 0 0 2 4 1 0 2 1 0 0 0 0 1 0     0 0 1 0 0 0 1 1 2 0 0 0 2 1 4 2 0 1 0 0 0 0 0 1     0 0 1 0 0 0 0 0 0 2 1 1 1 0 2 4 1 2 0 0 0 0 0 1     0 0 0 1 0 0 0 0 0 1 0 0 1 2 0 1 4 2 1 1 0 0 2 0     0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 2 2 4 0 0 1 1 0 2     1 1 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 4 2 2 1 1 0     0 0 1 1 0 2 0 0 0 0 1 0 0 0 0 0 1 0 2 4 1 0 2 1      0 0 0 0 1 0 1 1 0 0 2 0 0 0 0 0 0 1 2 1 4 2 0 1     0 0 0 0 1 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 2 4 1 2     0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 2 0 1 2 0 1 4 2    0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 2 0 1 1 2 2 4

  3. Permutations Card Shuffling Representation theory � � inc 4 ( τ − 1 σ ) Inc 4 , 4 =   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0      0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0      0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

  4. Permutations Card Shuffling Representation theory Motivation : Mysteries

  5. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! .

  6. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! . • these arose from a problem in computer science

  7. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! . • these arose from a problem in computer science • experimentation suggested some intriguing properties :

  8. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! . • these arose from a problem in computer science • experimentation suggested some intriguing properties : 1. Inc n,i Inc n,j = Inc n,j Inc n,i

  9. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! . • these arose from a problem in computer science • experimentation suggested some intriguing properties : 1. Inc n,i Inc n,j = Inc n,j Inc n,i 2. the eigenvalues are non-negative integers

  10. Permutations Card Shuffling Representation theory Motivation : Mysteries • for each n , we have a family of n matrices Inc n, 1 , Inc n, 2 , . . . , Inc n,n each of which is n ! × n ! . • these arose from a problem in computer science • experimentation suggested some intriguing properties : 1. Inc n,i Inc n,j = Inc n,j Inc n,i 2. the eigenvalues are non-negative integers • Questions : Is this true ? Why ? What are these integers ?

  11. Permutations Card Shuffling Representation theory Some Surprises

  12. Permutations Card Shuffling Representation theory Some Surprises they do commute ! ◦ 2011 : we gave an enumerative, inductive proof

  13. Permutations Card Shuffling Representation theory Some Surprises they do commute ! ◦ 2011 : we gave an enumerative, inductive proof eigenvalues ? ◦ 2013 : recent work with Ton Dieker : explicit formulas

  14. Permutations Card Shuffling Representation theory Some Surprises they do commute ! ◦ 2011 : we gave an enumerative, inductive proof eigenvalues ? ◦ 2013 : recent work with Ton Dieker : explicit formulas additional families of intriguing matrices : ◦ second family of matrices with similar properties (obtained by replacing inc k with another permutation statistic )

  15. Permutations Card Shuffling Representation theory Some Surprises they do commute ! ◦ 2011 : we gave an enumerative, inductive proof eigenvalues ? ◦ 2013 : recent work with Ton Dieker : explicit formulas additional families of intriguing matrices : ◦ second family of matrices with similar properties (obtained by replacing inc k with another permutation statistic ) connections with probability and representation theory : ◦ card shuffling and related random walks ◦ representation theory of the symmetric group

  16. Permutations Card Shuffling Representation theory Card Shuffling

  17. Permutations Card Shuffling Representation theory random-to-random shuffle deck of cards : σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 8 σ 9

  18. Permutations Card Shuffling Representation theory random-to-random shuffle deck of cards : σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 8 σ 9 remove a card at random : σ 3 σ 1 σ 2 ↑ σ 4 σ 5 σ 6 σ 7 σ 8 σ 9

  19. Permutations Card Shuffling Representation theory random-to-random shuffle deck of cards : σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 8 σ 9 remove a card at random : σ 3 σ 1 σ 2 ↑ σ 4 σ 5 σ 6 σ 7 σ 8 σ 9 insert the card at random : ↓ σ 1 σ 2 σ 4 σ 5 σ 6 σ 7 σ 3 σ 8 σ 9

  20. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   123   132     213       231     312     321

  21. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 123 9   132     213       231     312     321 3 ways to obtain 123 from 123 : 1 2 3 1 2 3 1 2 3

  22. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 2 123 9 9   132     213       231     312     321 2 ways to obtain 132 from 123 : 1 2 3 1 2 3

  23. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 2 2 123 9 9 9   132     213       231     312     321 2 ways to obtain 213 from 123 : 1 2 3 1 2 3

  24. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 2 2 1 123 9 9 9 9   132     213       231     312     321 1 way to obtain 231 from 123 : 1 2 3

  25. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 2 2 1 1 123 9 9 9 9 9   132     213       231     312     321 1 way to obtain 312 from 123 : 1 2 3

  26. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   3 2 2 1 1 0 123 9 9 9 9 9 9   132     213       231     312     321

  27. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   123 3 2 2 1 1 0   132 2 3 1 0 2 1     × 1 213 2 1 3 2 0 1     9   231 1 0 2 3 1 2     312 1 2 0 1 3 2     321 0 1 1 2 2 3

  28. Permutations Card Shuffling Representation theory Transition matrix of the random-to-random shuffle entries : probability of going from σ to τ using one shuffle 123 132 213 231 312 321   123 3 2 2 1 1 0   132 2 3 1 0 2 1     × 1 213 2 1 3 2 0 1     9   231 1 0 2 3 1 2     312 1 2 0 1 3 2     321 0 1 1 2 2 3 (renorm.) Inc n,n − 1 = random-to-random shuffle

  29. Permutations Card Shuffling Representation theory Properties of the transition matrix The transition matrix T governs properties of the random walk. typical questions ← → algebraic properties entries of T m ← → probability after m steps long-term behaviour ← → eigenvectors � π (limiting distribution) s.t. � π T = � π rate of convergence ← → governed by v T n − � → � π eigenvalues of T

  30. Permutations Card Shuffling Representation theory random walks on the chambers of a hyperplane arrangement

  31. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces :

  32. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : the origin

  33. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  34. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  35. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  36. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  37. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  38. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : rays emanating from the origin

  39. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  40. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  41. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  42. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  43. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  44. b Permutations Card Shuffling Representation theory faces of a hyperplane arrangement a set of hyperplanes partitions R n into faces : chambers cut out by the hyperplanes

  45. Permutations Card Shuffling Representation theory product of two faces � the face first encountered after a small xy := movement along a line from x toward y x y

  46. b b Permutations Card Shuffling Representation theory product of two faces � the face first encountered after a small xy := movement along a line from x toward y x y

  47. b b Permutations Card Shuffling Representation theory product of two faces � the face first encountered after a small xy := movement along a line from x toward y x y

  48. b b Permutations Card Shuffling Representation theory product of two faces � the face first encountered after a small xy := movement along a line from x toward y x xy y

  49. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement v ∈ R n : v i = v j } H i,j = { � hyperplanes :

  50. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement v ∈ R n : v i = v j } H i,j = { � hyperplanes : • consider a vector � v that belongs to a chamber

  51. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement v ∈ R n : v i = v j } H i,j = { � hyperplanes : • consider a vector � v that belongs to a chamber • we can order the entries of � v in increasing order ; e.g. : v 5 < v 1 < v 3 < v 2 < v 4

  52. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement v ∈ R n : v i = v j } H i,j = { � hyperplanes : • consider a vector � v that belongs to a chamber • we can order the entries of � v in increasing order ; e.g. : v 5 < v 1 < v 3 < v 2 < v 4 • so chambers correspond to permutations : � � v 5 < v 1 < v 3 < v 2 < v 4 ← → 5 , 1 , 3 , 2 , 4

  53. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement v ∈ R n : v i = v j } H i,j = { � hyperplanes : • consider a vector � v that belongs to a chamber • we can order the entries of � v in increasing order ; e.g. : v 5 < v 1 < v 3 < v 2 < v 4 • so chambers correspond to permutations : � � v 5 < v 1 < v 3 < v 2 < v 4 ← → 5 , 1 , 3 , 2 , 4 • if � v lies on H i,j , then v i < v j becomes v i = v j : � � v 1 = v 5 < v 2 = v 3 < v 4 ← → { 1 , 5 } , { 2 , 3 } , { 4 }

  54. Permutations Card Shuffling Representation theory Special Case : The “Braid” Arrangement combinatorial description : faces ↔ ordered set partitions of { 1 , . . . , n } : � � � � { 2 , 3 } , { 4 } , { 1 , 5 } { 4 } , { 1 , 5 } , { 2 , 3 } � = chambers ↔ partitions into singletons � � { 2 } , { 3 } , { 4 } , { 1 } , { 5 } product ↔ intersection of sets in the partition

  55. Permutations Card Shuffling Representation theory Product of set compositions

  56. Permutations Card Shuffling Representation theory Product of set compositions � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 }{ 2 }

  57. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 }{ 2 } � { 2 , 5 } ∩ { 4 } =

  58. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 }{ 2 } � ∅ =

  59. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 }{ 2 } � =

  60. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 } { 1 }{ 5 }{ 6 }{ 3 }{ 2 } � { 2 , 5 } ∩ { 1 } =

  61. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 } { 1 }{ 5 }{ 6 }{ 3 }{ 2 } � =

  62. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 } { 5 }{ 6 }{ 3 }{ 2 } � { 2 , 5 } ∩ { 5 } =

  63. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 } { 5 }{ 6 }{ 3 }{ 2 } � { 5 } =

  64. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 } { 5 }{ 6 }{ 3 }{ 2 } � { 5 } =

  65. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 } { 6 }{ 3 }{ 2 } � { 5 }{ 2 , 5 } ∩ { 6 } =

  66. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 } { 6 }{ 3 }{ 2 } � { 5 } =

  67. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 } { 3 }{ 2 } � { 5 }{ 2 , 5 } ∩ { 3 } =

  68. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 } { 2 } � { 5 }{ 2 , 5 } ∩ { 2 } =

  69. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 }{ 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 } { 2 } � { 5 }{ 2 } =

  70. Permutations Card Shuffling Representation theory Product of set compositions � � � � � � � � { 2 , 5 } { 1 , 3 , 4 , 6 } · { 4 }{ 1 }{ 5 }{ 6 }{ 3 }{ 2 } � { 5 }{ 2 }{ 1 , 3 , 4 , 6 } ∩ { 4 } =

Recommend

Shuffling properties for products of random permutations Olivier Bernardi (MIT) Joint work with

Shuffling properties for products of random permutations Olivier Bernardi (MIT) Joint work with

Shuffling properties for products of random permutations Olivier Bernardi (MIT) Joint work with Rosena Du, Alejandro Morales, Richard Stanley 3 9 7 10 2 1 8 5 11 4 6 Halifax, June 2012 Warm up Question: Let be a uniformly random

592 views • 48 slides
Mixing Times of Self-Organizing Lists Applications and Biased Permutations Previous Work New

Mixing Times of Self-Organizing Lists Applications and Biased Permutations Previous Work New

Card Shuffling Mixing Times of Self-Organizing Lists Applications and Biased Permutations Previous Work New Results Amanda Pascoe Streib NIST Applied and Computational Math Division Joint work with: Prateek Bhakta, Sarah Miracle, Dana

1.67k views • 118 slides
Eigenvalues of symmetrized shuffling operators Nadia Lafrenire Universit du Qubec

Eigenvalues of symmetrized shuffling operators Nadia Lafrenire Universit du Qubec

Eigenvalues of symmetrized shuffling operators Nadia Lafrenire Universit du Qubec Montral FPSAC 2019 Pick a cardany card! Pick a cardany card! Pick a cardany card! Pick a cardany card! The random-to-random shuffle The

562 views • 30 slides
Finger and Palm Interactions in a Multi- Touch Card Game Christian Reichart Course: Interaction

Finger and Palm Interactions in a Multi- Touch Card Game Christian Reichart Course: Interaction

Finger and Palm Interactions in a Multi- Touch Card Game Christian Reichart Course: Interaction Engineering Winter semester 2014/15 Instructor: Prof. Dr. Michael Kipp Real Card Games - Shuffling - Dealing out cards - Drawing - Playing out

342 views • 17 slides
Representation Theory and Holomorphic Polydifferentials Adam Wood Department of Mathematics

Representation Theory and Holomorphic Polydifferentials Adam Wood Department of Mathematics

Representation Theory and Holomorphic Polydifferentials Adam Wood Department of Mathematics University of Iowa Research Seminar, St. Olaf College April 26, 2019 Outline Introduction to (Modular) Representation Theory Representation Theory

1.39k views • 114 slides
Groups, Representation Theory, and Curves Adam Wood Department of Mathematics University of Iowa

Groups, Representation Theory, and Curves Adam Wood Department of Mathematics University of Iowa

Groups, Representation Theory, and Curves Adam Wood Department of Mathematics University of Iowa MAA MathFest Great Talks for a General Audience August 3, 2019 Outline Introduction to (Modular) Representation Theory Representation Theory of

955 views • 81 slides
On the Shuffling Algorithm for the Aztec Nordenstam eno@kth.se Diamond Background Shuffling

On the Shuffling Algorithm for the Aztec Nordenstam eno@kth.se Diamond Background Shuffling

On the Shuffling Algorithm for the Aztec Diamond Eric On the Shuffling Algorithm for the Aztec Nordenstam eno@kth.se Diamond Background Shuffling algorithm Eric Nordenstam Warrens Process eno@kth.se Aztec diamond point

672 views • 44 slides
CSCI 246 Class 20 PROBABILITIES, PERMUTATIONS, COMBINATIONS PART 2 Quiz Questions

CSCI 246 Class 20 PROBABILITIES, PERMUTATIONS, COMBINATIONS PART 2 Quiz Questions

CSCI 246 Class 20 PROBABILITIES, PERMUTATIONS, COMBINATIONS PART 2 Quiz Questions Lecture 34: How many possible 5 card poker hands are there? Notes and Clarifications No Office Hours Today/Tomorrow Reminder: Quiz due

354 views • 6 slides
Shuffling Cards via One-sided Transpositions Stephen Connor Joint work with Oliver Matheau-Raven

Shuffling Cards via One-sided Transpositions Stephen Connor Joint work with Oliver Matheau-Raven

Shuffling Cards via One-sided Transpositions Stephen Connor Joint work with Oliver Matheau-Raven and Michael Bate SAMBa Conference July 2020 Introduction Consider the following method for shuffling a pack of n cards: Right hand chooses a

423 views • 16 slides
Number Theory and Representation Theory A conference in honor of the 60th birthday of Benedict

Number Theory and Representation Theory A conference in honor of the 60th birthday of Benedict

Number Theory and Representation Theory A conference in honor of the 60th birthday of Benedict Gross Harvard University, Cambridge June 2010 Elliptic curves over real quadratic fields, and the Birch and Swinnerton-Dyer conjecture ... A

1.59k views • 103 slides
CS 103: Representation Learning, Information Theory and Control Lecture 5, Feb 8, 2019

CS 103: Representation Learning, Information Theory and Control Lecture 5, Feb 8, 2019

CS 103: Representation Learning, Information Theory and Control Lecture 5, Feb 8, 2019 Representation Learning and Information Bottleneck Desiderata for representations An optimal representation z of the data x for the task y is a stochastic

279 views • 8 slides
Local convergence for random permutations The case of uniform pattern-avoiding permutations

Local convergence for random permutations The case of uniform pattern-avoiding permutations

Local convergence for random permutations The case of uniform pattern-avoiding permutations Jacopo Borga, Institut fr Mathematik, Universitt Zrich July 9, 2018 Dartmouth College, Hanover, New Hampshire Our goal 1. Scaling limits: Our

2.18k views • 175 slides
JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by

JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by

JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by A.J.Hobson 19.2.1 Introduction 19.2.2 Rules of permutations and combinations 19.2.3 Permutations of sets with some objects alike UNIT 19.2 -

400 views • 11 slides
Representing permutations without permutations The expressive power of sequential intersection

Representing permutations without permutations The expressive power of sequential intersection

Representing permutations without permutations The expressive power of sequential intersection Pierre VIAL Equipe Gallinette Inria (LS2N CNRS) June 10, 2018 Non-idempotent typing P. Vial 0 1 /33 Outline Non-idempotent Rigid vs.

1.4k views • 123 slides
Applications of the Defect to Representation Theory Jeremy Russell The College of New Jersey

Applications of the Defect to Representation Theory Jeremy Russell The College of New Jersey

Applications of the Defect to Representation Theory Jeremy Russell The College of New Jersey Representation Theory Seminar January 23, 2015 Goals of The Talk 1 Recall Auslanders construction of the defect w : fp ( A , Ab ) A Goals

2.2k views • 172 slides
Between lattice theory and representation theory David E Speyer Based on results of Laurent

Between lattice theory and representation theory David E Speyer Based on results of Laurent

Between lattice theory and representation theory David E Speyer Based on results of Laurent Demonet, Colin Ingalls, Osamu Iyama, Nathan Reading, Idun Reiten, David E Speyer and Hugh Thomas. 1 Agenda 1. Coxeter groups and Cambrian lattices 2.

870 views • 31 slides
Introduction to Complexity Theory Winter 2005/06 Dr. Axel Gromann Knowledge Representation and

Introduction to Complexity Theory Winter 2005/06 Dr. Axel Gromann Knowledge Representation and

Introduction to Complexity Theory Winter 2005/06 Dr. Axel Gromann Knowledge Representation and Reasoning Group Artificial Intelligence Institute 1 Computational complexity theory Complexity theory is part of the theory of computation dealing

915 views • 15 slides
Purchasing Card Program Training for Cardholders Introduction to the P-Card Program What is a

Purchasing Card Program Training for Cardholders Introduction to the P-Card Program What is a

Purchasing Card Program Training for Cardholders Introduction to the P-Card Program What is a P-Card? VISA credit card issued by Bank of America Used to make purchase of goods and some services of $2,500 or less P-Card Controls

480 views • 34 slides
Trees, Strings, and Representation Theory Adam Wood Department of Mathematics University of Iowa

Trees, Strings, and Representation Theory Adam Wood Department of Mathematics University of Iowa

Trees, Strings, and Representation Theory Adam Wood Department of Mathematics University of Iowa Graduate and Undergraduate Student Seminar University of Iowa October 9, 2019 Outline Motivating Example Representation Theory of Finite Groups

1.4k views • 94 slides
Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents.

Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents.

Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents. Mathilde Bouvel Elisa Pergola GASCom 2008 liafa Definitions Duplication-Loss Characterization Posets Enumeration Conclusion Outline of the talk 1

620 views • 31 slides
Open problem session Representation Theory XVI Dubrovnik June 28, 2019 July 5, 2019 1

Open problem session Representation Theory XVI Dubrovnik June 28, 2019 July 5, 2019 1

Open problem session Representation Theory XVI Dubrovnik June 28, 2019 July 5, 2019 1 Introduction These are open problems presented at the last meeting of the Lie Groups Section at the conference Representation Theory XVI held at the

434 views • 9 slides
A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac

A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac

A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac Ali ahin March 6, 2017 Motivation 2/14 Permutations are main nonlinear part of symmetric primitives Quadratic permutations can be used to

381 views • 22 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 &gt; a 2 &lt; a 3 &gt; a 4 &lt; , and reverse alternating if

1.14k views • 98 slides
Training for Cardholders Introduction to the P-Card Program What is a P-Card? VISA credit

Training for Cardholders Introduction to the P-Card Program What is a P-Card? VISA credit

Purchasing Card Program Training for Cardholders Introduction to the P-Card Program What is a P-Card? VISA credit card issued by Bank of America Used to make purchase of goods and some services of $2,500 or less P-Card Controls

933 views • 26 slides

More recommend