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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 } =

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