Unimodality of q -Eulerian polynomials and q , p -Eulerian - PowerPoint PPT Presentation

Unimodality of q -Eulerian polynomials and q , p -Eulerian polynomials Michelle Wachs University of Miami Joint work with John Shareshian and with Anthony Henderson

Eulerian numbers and Eulerian polynomials Eulerian numbers: a n , j := |{ σ ∈ S n : des ( σ ) = j }| = |{ σ ∈ S n : exc ( σ ) = j }| , des ( σ ) = { i ∈ [ n − 1] : σ ( i ) > σ ( i + 1) } exc ( σ ) = { i ∈ [ n − 1] : σ ( i ) > i } Eulerian polynomials: n − 1 a n , j t j = t des ( σ ) = � � � t exc ( σ ) A n ( t ) := j =0 σ ∈ S n σ ∈ S n

Eulerian numbers and Eulerian polynomials Eulerian numbers: a n , j := |{ σ ∈ S n : des ( σ ) = j }| = |{ σ ∈ S n : exc ( σ ) = j }| , des ( σ ) = { i ∈ [ n − 1] : σ ( i ) > σ ( i + 1) } exc ( σ ) = { i ∈ [ n − 1] : σ ( i ) > i } Eulerian polynomials: n − 1 a n , j t j = t des ( σ ) = � � � t exc ( σ ) A n ( t ) := j =0 σ ∈ S n σ ∈ S n Euler’s exponential generating function formula: A n ( t ) z n 1 − t � n ! = e z ( t − 1) − t n ≥ 0

Eulerian numbers and Eulerian polynomials Eulerian numbers: a n , j := |{ σ ∈ S n : des ( σ ) = j }| = |{ σ ∈ S n : exc ( σ ) = j }| , des ( σ ) = { i ∈ [ n − 1] : σ ( i ) > σ ( i + 1) } exc ( σ ) = { i ∈ [ n − 1] : σ ( i ) > i } Eulerian polynomials: n − 1 a n , j t j = t des ( σ ) = � � � t exc ( σ ) A n ( t ) := j =0 σ ∈ S n σ ∈ S n Euler’s exponential generating function formula: A n ( t ) z n e z ( t − 1) − t = (1 − t ) e t 1 − t � n ! = e tz − te z n ≥ 0

Symmetry and Unimodality Eulerian numbers a n , j n \ j 0 1 2 3 4 1 1 2 1 1 3 1 4 1 4 1 11 11 1 5 1 26 66 26 1

Symmetry and Unimodality Eulerian numbers a n , j n \ j 0 1 2 3 4 1 1 2 1 1 3 1 4 1 4 1 11 11 1 5 1 26 66 26 1 A stronger property: γ -positivity ⌊ n − 1 2 ⌋ � γ n , i t i (1 + t ) n − 1 − 2 i , A n ( t ) = γ n , i ∈ N i =0 Foata & Sh¨ utzenberger (1970): γ n , i = |{ σ ∈ S n | σ 0 has no double descents & des ( σ ) = i }|

Geometric interpretation ( a n , 0 , a n , 1 , . . . , a n , n − 1 ) is the h-vector of the barycentric subdivision of the ( n − 1)-simplex (type A Coxeter complex). Stanley (1980): The h -vector of every simplicial polytope is unimodal (and symmetric). The γ vector of a d -dimensional simplicial polytope ∆ is defined by ⌊ d 2 ⌋ d h i (∆) t i = � � γ i (∆) t i (1 + t ) n − 1 − 2 i i =0 i =0

Geometric interpretation ( a n , 0 , a n , 1 , . . . , a n , n − 1 ) is the h-vector of the barycentric subdivision of the ( n − 1)-simplex (type A Coxeter complex). Stanley (1980): The h -vector of every simplicial polytope is unimodal (and symmetric). The γ vector of a d -dimensional simplicial polytope ∆ is defined by ⌊ d 2 ⌋ d h i (∆) t i = � � γ i (∆) t i (1 + t ) n − 1 − 2 i i =0 i =0 Gal’s Conjecture (2005): The γ vector of every flag homology sphere is nonnegative, i.e. γ i (∆) ≥ 0 for all i .

Geometric interpretation ( a n , 0 , a n , 1 , . . . , a n , n − 1 ) is the h-vector of the barycentric subdivision of the ( n − 1)-simplex (type A Coxeter complex). Stanley (1980): The h -vector of every simplicial polytope is unimodal (and symmetric). The γ vector of a d -dimensional simplicial polytope ∆ is defined by ⌊ d 2 ⌋ d h i (∆) t i = � � γ i (∆) t i (1 + t ) n − 1 − 2 i i =0 i =0 Gal’s Conjecture (2005): The γ vector of every flag homology sphere is nonnegative, i.e. γ i (∆) ≥ 0 for all i . Peterson, Stembridge: true for all Coxeter complexes

q -Eulerian numbers and q -Eulerian polynomials q -Eulerian numbers � q maj ( σ ) − j a n , j ( q ) := σ ∈ S n exc ( σ ) = j q -Eulerian polynomials: n − 1 a n , j ( q ) t j = � � q maj ( σ ) − exc ( σ ) t exc ( σ ) A n ( q , t ) := j =0 σ ∈ S n

q -Eulerian numbers and q -Eulerian polynomials q -Eulerian numbers � q maj ( σ ) − j a n , j ( q ) := σ ∈ S n exc ( σ ) = j q -Eulerian polynomials: n − 1 a n , j ( q ) t j = � � q maj ( σ ) − exc ( σ ) t exc ( σ ) A n ( q , t ) := j =0 σ ∈ S n Shareshian and MW (2005): A n ( q , t ) z n (1 − t ) exp q ( z ) � = [ n ] q ! exp q ( tz ) − t exp q ( z ) n ≥ 0 Proof: We specialize a symmetric function analog involving the Eulerian quasisymmetric functions Q n , j .

Symmetry and Unimodality of A n ( q , t ) n \ j 0 1 2 3 4 1 1 2 1 1 2 + q + q 2 3 1 1 3 + 2 q + 3 q 2 + 2 q 3 + q 4 3 + 2 q + 3 q 2 + 2 q 3 + q 4 4 1 1 4 + 3 q + 5 q 2 + ... 6 + 6 q + 11 q 2 + ... 4 + 3 q + 5 q 2 + ... 5 1 1

Symmetry and Unimodality of A n ( q , t ) n \ j 0 1 2 3 4 1 1 2 1 1 2 + q + q 2 3 1 1 3 + 2 q + 3 q 2 + 2 q 3 + q 4 3 + 2 q + 3 q 2 + 2 q 3 + q 4 4 1 1 4 + 3 q + 5 q 2 + ... 6 + 6 q + 11 q 2 + ... 4 + 3 q + 5 q 2 + ... 5 1 1 ⌊ n − 1 2 ⌋ � γ n , i ( q ) t i (1 + t ) n − 1 − 2 i A n ( q , t ) = i =0

Symmetry and Unimodality of A n ( q , t ) n \ j 0 1 2 3 4 1 1 2 1 1 2 + q + q 2 3 1 1 3 + 2 q + 3 q 2 + 2 q 3 + q 4 3 + 2 q + 3 q 2 + 2 q 3 + q 4 4 1 1 4 + 3 q + 5 q 2 + ... 6 + 6 q + 11 q 2 + ... 4 + 3 q + 5 q 2 + ... 5 1 1 ⌊ n − 1 2 ⌋ � γ n , i ( q ) t i (1 + t ) n − 1 − 2 i A n ( q , t ) = i =0 Shareshian and MW: q maj ( σ − 1 ) ∈ N [ q ] � γ n , i ( q ) = σ ∈ DD n , i where DD n , i := { σ ∈ S n | σ 0 has no double descents & des ( σ ) = i }

Eulerian quasisymmetric functions - Shareshian and MW For σ ∈ S n , let ¯ σ be obtained by placing bars above each excedance. ¯ 5¯ 314¯ 62 View ¯ σ as a word over ordered alphabet { ¯ 1 < ¯ 2 < · · · < ¯ n < 1 < 2 < · · · < n } . Define DEX ( σ ) := DES (¯ σ ) DEX (531462) = DES (¯ 5 . ¯ 314 . ¯ 62) = { 1 , 4 }

Eulerian quasisymmetric functions - Shareshian and MW For σ ∈ S n , let ¯ σ be obtained by placing bars above each excedance. ¯ 5¯ 314¯ 62 View ¯ σ as a word over ordered alphabet { ¯ 1 < ¯ 2 < · · · < ¯ n < 1 < 2 < · · · < n } . Define DEX ( σ ) := DES (¯ σ ) DEX (531462) = DES (¯ 5 . ¯ 314 . ¯ 62) = { 1 , 4 } We prove � i ∈ DEX ( σ ) i = maj ( σ ) − exc ( σ ).

Eulerian quasisymmetric functions - Shareshian and MW For σ ∈ S n , let ¯ σ be obtained by placing bars above each excedance. ¯ 5¯ 314¯ 62 View ¯ σ as a word over ordered alphabet { ¯ 1 < ¯ 2 < · · · < ¯ n < 1 < 2 < · · · < n } . Define DEX ( σ ) := DES (¯ σ ) DEX (531462) = DES (¯ 5 . ¯ 314 . ¯ 62) = { 1 , 4 } We prove � i ∈ DEX ( σ ) i = maj ( σ ) − exc ( σ ). maj (5 . 3 . 146 . 2) − exc (531462) = 8 − 3 = 5

Eulerian quasisymmetric functions - Shareshian and MW (2005) For all j ∈ { 0 , 1 , . . . , n − 1 } , define the Eulerian quasisymmetric function � Q n , j := F DEX ( σ ) , σ ∈ S n exc ( σ ) = j where for S ⊆ [ n − 1], � F S := F S ( x 1 , x 2 , . . . ) := x i 1 x i 2 . . . x i n i 1 ≥ i 2 ≥ · · · ≥ i n i k > i k +1 ∀ k ∈ S

Eulerian quasisymmetric functions - Shareshian and MW (2005) For all j ∈ { 0 , 1 , . . . , n − 1 } , define the Eulerian quasisymmetric function � Q n , j := F DEX ( σ ) , σ ∈ S n exc ( σ ) = j where for S ⊆ [ n − 1], � F S := F S ( x 1 , x 2 , . . . ) := x i 1 x i 2 . . . x i n i 1 ≥ i 2 ≥ · · · ≥ i n i k > i k +1 ∀ k ∈ S Stable principal specialization: p s ( F S ) := F S ( q 0 , q 1 , q 2 , . . . ) = ( q ; q ) − 1 P i ∈ S i n q where ( p ; q ) n := (1 − p )(1 − pq ) . . . (1 − pq n − 1 )

Symmetric function analog of Euler’s formula � p s ( Q n , j ) = ( q ; q ) − 1 q maj ( σ ) − exc ( σ ) n σ ∈ S n exc ( σ ) = j

Symmetric function analog of Euler’s formula � p s ( Q n , j ) = ( q ; q ) − 1 q maj ( σ ) − exc ( σ ) n σ ∈ S n exc ( σ ) = j Shareshian and MW (2006): n − 1 (1 − t ) H ( z ) Q n , j t j z n = � � H ( tz ) − tH ( z ) , n ≥ 0 j =0 n ≥ 0 h n z n . where H ( z ) = �

Symmetric function analog of Euler’s formula � p s ( Q n , j ) = ( q ; q ) − 1 q maj ( σ ) − exc ( σ ) n σ ∈ S n exc ( σ ) = j Shareshian and MW (2006): n − 1 (1 − t ) H ( z ) Q n , j t j z n = � � H ( tz ) − tH ( z ) , n ≥ 0 j =0 n ≥ 0 h n z n . where H ( z ) = � x i �→ q i − 1 and z �→ z (1 − q ) = ⇒ A n ( q , t ) z n (1 − t ) exp q ( z ) � = [ n ] q ! exp q ( tz ) − t exp q ( z ) n ≥ 0

Another occurrence of this symmetric function Gessel: (1 − t ) H ( z ) x w t i (1 + t ) n − 1 − 2 i = � � z n 1 + H ( tz ) − tH ( z ) n ≥ 1 w ∈ DD n , i ( P ) where x w := x w 1 x w 2 . . . x w n and DD n , i ( P ) := { w ∈ P n | w 0 has no double descents & des ( w ) = i } 779 . 1558 . 25 ∈ DD 9 , 2 ( P )

Another occurrence of this symmetric function Gessel: (1 − t ) H ( z ) x w t i (1 + t ) n − 1 − 2 i = � � z n 1 + H ( tz ) − tH ( z ) n ≥ 1 w ∈ DD n , i ( P ) where x w := x w 1 x w 2 . . . x w n and DD n , i ( P ) := { w ∈ P n | w 0 has no double descents & des ( w ) = i } 779 . 1558 . 25 ∈ DD 9 , 2 ( P ) 2 5 1 5 5 8 7 7 9