EulerMahonian Statistics Via Polyhedral Geometry [ n ] q ! n ! - PowerPoint PPT Presentation

Euler–Mahonian Statistics Via Polyhedral Geometry [ n ] q ! n ! Matthias Beck San Francisco State University Benjamin Braun University of Kentucky arXiv:1109.3353 Adv. Math. (2013)

Permutation Statistics π ∈ S n — permutation of { 1 , 2 , . . . , n } Goal: study certain statistics of S n (and other reflection groups), e.g., � � Des( π ) := j : π ( j ) > π ( j + 1) des( π ) := # Des( π ) � maj( π ) := j j ∈ Des( π ) � � inv( π ) := # ( j, k ) : j < k and π ( j ) > π ( k ) Example S 3 = { [123] , [213] , [132] , [312] , [132] , [321] } t des( π ) = 1 + 4 t + t 2 � π ∈ S 3 Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 2

Permutation Statistics π ∈ S n — permutation of { 1 , 2 , . . . , n } Goal: study certain statistics of S n (and other reflection groups), e.g., � � Des( π ) := j : π ( j ) > π ( j + 1) des( π ) := # Des( π ) � maj( π ) := j j ∈ Des( π ) � � inv( π ) := # ( j, k ) : j < k and π ( j ) > π ( k ) More generally, for a Coxeter group W with generators s 1 , s 2 , . . . , s n − 1 , the (right) descent set of σ ∈ W is Des( σ ) := { j : l ( σs j ) < l ( σ ) } Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 2

Permutation Statistics π ∈ S n — permutation of { 1 , 2 , . . . , n } Goal: study certain statistics of S n (and other reflection groups), e.g., � � Des( π ) := j : π ( j ) > π ( j + 1) des( π ) := # Des( π ) � maj( π ) := j j ∈ Des( π ) � � inv( π ) := # ( j, k ) : j < k and π ( j ) > π ( k ) π ∈ S n t des( π ) � ( k + 1) n t k = � Sample Theorem 1 [Euler] (1 − t ) n +1 k ≥ 0 t inv( π ) = � � t maj( π ) Sample Theorem 2 [MacMahon] π ∈ S n π ∈ S n Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 2

Permutation Statistics [Euler] [MacMahon] π ∈ S n t des( π ) � ( k +1) n t k = t inv( π ) = � � � t maj( π ) (1 − t ) n +1 k ≥ 0 π ∈ S n π ∈ S n Goal: new identities of this kind π ∈ S n t des( π ) q maj( π ) � q t k = � [ k +1] n Sample Theorem 3 [MacMahon] � n j =0 (1 − tq j ) k ≥ 0 Sample Theorem 4 [Brenti, Steingr´ ımsson] ( π,ǫ ) ∈ B n t des( π,ǫ ) � (2 k + 1) n t k = � (1 − t ) n +1 k ≥ 0 ( π, ǫ ) — signed permutation with π ∈ S n and ǫ ∈ {± 1 } Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 3

Permutation Statistics [Euler] [MacMahon] π ∈ S n t des( π ) � ( k +1) n t k = t inv( π ) = � � � t maj( π ) (1 − t ) n +1 k ≥ 0 π ∈ S n π ∈ S n Goal: new identities of this kind π ∈ S n t des( π ) q maj( π ) � q t k = � [ k +1] n Sample Theorem 3 [MacMahon] � n j =0 (1 − tq j ) k ≥ 0 Sample Theorem 5 [Chow–Gessel] ( π,ǫ ) ∈ B n s neg( ǫ ) t des( π,ǫ ) q maj( π,ǫ ) � ([ k + 1] q + s [ k ] q ) n t k = � � n j =0 (1 − tq j ) k ≥ 0 ( π, ǫ ) — signed permutation with π ∈ S n and ǫ ∈ {± 1 } Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 3

Permutation Statistics [Euler] [MacMahon] π ∈ S n t des( π ) � ( k +1) n t k = t inv( π ) = � � � t maj( π ) (1 − t ) n +1 k ≥ 0 π ∈ S n π ∈ S n Goal: new identities of this kind bijective proofs (integer partitions) ◮ Coxeter groups (invariant theory) ◮ symmetric & quasisymmetric functions ◮ polyhedral geometry ◮ Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 3

Enter Geometry [Euler] [MacMahon] π ∈ S n t des( π ) π ∈ S n t des( π ) q maj( π ) � � ( k + 1) n t k = q t k = � � [ k + 1] n � n (1 − t ) n +1 j =0 (1 − tq j ) k ≥ 0 k ≥ 0 # ( k [0 , 1] n ∩ Z n ) = ( k + 1) n is the Ehrhart polynomial of the unit n -cube Use braid arrangement { x j = x k : 1 ≤ j < k ≤ n } triangulation of [0 , 1] n : � � [0 , 1] n = x ∈ R n : 0 ≤ x π ( n ) ≤ x π ( n − 1) ≤ · · · ≤ x π (1) ≤ 1 � π ∈ S n Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 4

Enter Geometry [Euler] [MacMahon] π ∈ S n t des( π ) π ∈ S n t des( π ) q maj( π ) � � ( k + 1) n t k = q t k = � � [ k + 1] n � n (1 − t ) n +1 j =0 (1 − tq j ) k ≥ 0 k ≥ 0 # ( k [0 , 1] n ∩ Z n ) = ( k + 1) n is the Ehrhart polynomial of the unit n -cube Use braid arrangement { x j = x k : 1 ≤ j < k ≤ n } triangulation of [0 , 1] n : � � x ∈ R n : 0 ≤ x π ( n ) ≤ x π ( n − 1) ≤ · · · ≤ x π (1) ≤ 1 , [0 , 1] n = � x π ( j +1) < x π ( j ) if j ∈ Des( π ) π ∈ S n Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 4

Enter Geometry [Euler] [MacMahon] π ∈ S n t des( π ) π ∈ S n t des( π ) q maj( π ) � � ( k + 1) n t k = q t k = � � [ k + 1] n � n (1 − t ) n +1 j =0 (1 − tq j ) k ≥ 0 k ≥ 0 # ( k [0 , 1] n ∩ Z n ) = ( k + 1) n is the Ehrhart polynomial of the unit n -cube � � x ∈ R n : 0 ≤ x π ( n ) ≤ x π ( n − 1) ≤ · · · ≤ x π (1) ≤ 1 , [0 , 1] n = � x π ( j +1) < x π ( j ) if j ∈ Des( π ) π ∈ S n Each simplex on the right is unimodular with Ehrhart series t #[ strict inequalities ] (1 − t ) n +1 t des( π ) ( k + 1) n t k = � � = ⇒ (1 − t ) n +1 k ≥ 0 π ∈ S n Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 4

More Geometry [Euler] [MacMahon] π ∈ S n t des( π ) π ∈ S n t des( π ) q maj( π ) � � ( k + 1) n t k = q t k = � � [ k + 1] n � n (1 − t ) n +1 j =0 (1 − tq j ) k ≥ 0 k ≥ 0 � � x ∈ R n : 0 ≤ x π ( n ) ≤ x π ( n − 1) ≤ · · · ≤ x π (1) ≤ 1 , [0 , 1] n = � x π ( j +1) < x π ( j ) if j ∈ Des( π ) π ∈ S n For P ⊂ R n consider σ cone( P ) ( z 0 , z 1 , . . . , z n ) := � z m 0 0 z m 1 · · · z m n 1 n m ∈ cone( P ) ∩ Z n +1 n � � 1 + z j + z 2 j + · · · + z k z k � � σ cone([0 , 1] n ) ( z 0 , z 1 , . . . , z n ) = 0 j j =1 k ≥ 0 n � � [ k + 1] z j z k = 0 k ≥ 0 j =1 Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 5

More Geometry [Euler] [MacMahon] π ∈ S n t des( π ) π ∈ S n t des( π ) q maj( π ) � � ( k + 1) n t k = q t k = � � [ k + 1] n � n (1 − t ) n +1 j =0 (1 − tq j ) k ≥ 0 k ≥ 0 � � x ∈ R n : 0 ≤ x π ( n ) ≤ x π ( n − 1) ≤ · · · ≤ x π (1) ≤ 1 , [0 , 1] n = � x π ( j +1) < x π ( j ) if j ∈ Des( π ) π ∈ S n n � j ∈ Des( π ) z 0 z π (1) z π (2) · · · z π ( j ) � � � [ k + 1] z j z k Theorem 0 = � n � � 1 − z 0 z π (1) z π (2) · · · z π ( j ) j =0 j =1 k ≥ 0 π ∈ S n Remark Philosophy very close to that of P -partitions MacMahon’s theorem follows by setting z 0 = t and z 1 = z 2 = · · · = z n = q Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 5

Type-B Permutation Statistics ( π, ǫ ) — signed permutation with π ∈ S n and ǫ ∈ {± 1 } Use the natural decent statistics � � Des( π ) := j : ǫ j π ( j ) > ǫ j +1 π ( j + 1) [ ǫ 0 π (0) := 0] des( π ) := # Des( π ) � maj( π ) := j j ∈ Des( π ) Sample Theorem 4 [Brenti, Steingr´ ımsson] ( π,ǫ ) ∈ B n t des( π,ǫ ) � (2 k + 1) n t k = � (1 − t ) n +1 k ≥ 0 Sample Theorem 5 [Chow–Gessel] ( π,ǫ ) ∈ B n s neg( ǫ ) t des( π,ǫ ) q maj( π,ǫ ) � ([ k + 1] q + s [ k ] q ) n t k = � � n j =0 (1 − tq j ) k ≥ 0 Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 6

Type-B Geometry [Brenti, Steingr´ ımsson] ( π,ǫ ) ∈ B n t des( π,ǫ ) � (2 k + 1) n t k = � (1 − t ) n +1 k ≥ 0 Use the type-B arrangement { x j = ± x k , x j = 0 : 1 ≤ j < k ≤ n } to triangulate [ − 1 , 1] n : � � x ∈ R n : 0 ≤ ǫ n x π ( n ) ≤ ǫ n − 1 x π ( n − 1) ≤ · · · ≤ ǫ 1 x π (1) ≤ 1 [ − 1 , 1] n = � ǫ j +1 x π ( j +1) < ǫ j x π ( j ) if j ∈ Des( π, ǫ ) ( π,ǫ ) ∈ B n Each simplex on the right is unimodular with Ehrhart series t #[ strict inequalities ] (1 − t ) n +1 Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 7

More Type-B Geometry [Chow–Gessel] ( π,ǫ ) ∈ B n s neg( ǫ ) t des( π,ǫ ) q maj( π,ǫ ) � ([ k + 1] q + s [ k ] q ) n t k = � � n j =0 (1 − tq j ) k ≥ 0 � � x ∈ R n : 0 ≤ ǫ n x π ( n ) ≤ ǫ n − 1 x π ( n − 1) ≤ · · · ≤ ǫ 1 x π (1) ≤ 1 [ − 1 , 1] n = � ǫ j +1 x π ( j +1) < ǫ j x π ( j ) if j ∈ Des( π, ǫ ) ( π,ǫ ) ∈ B n n � � � � w j [ k + 1] z j + w − j z − 1 z k Theorem − j [ k ] z − 1 0 = − j j =1 k ≥ 0 ǫ j � z 0 z ǫ 1 ǫ 1 π (1) z ǫ 2 ǫ 2 π (2) · · · z ǫ j π ( j ) j ∈ Des( π,ǫ ) � � � z − 1 w j − j w − j n � � ǫ j � 1 − z 0 z ǫ 1 ǫ 1 π (1) z ǫ 2 ǫ j =1 ǫ j = − 1 ( π,ǫ ) ∈ B n ǫ 2 π (2) · · · z ǫ j π ( j ) j =0 Chow–Gessel’s theorem follows with z 0 = t , z 1 = · · · = z n = z − 1 − 1 = · · · = z − 1 − n = q , w − 1 = · · · = w − n = s , and w 1 = · · · = w n = 1 Euler–Mahonian Statistics Via Polyhedral Geometry Matthias Beck 8