Generalized Inversion Sequences Carla D. Savage Department of - PowerPoint PPT Presentation

Generalized Inversion Sequences Carla D. Savage Department of Computer Science North Carolina State University CanaDAM 2013 Memorial University of Newfoundland, June 10, 2013

Permutations and Descents S n : set of permutations π : { 1 , 2 , . . . , n } → { 1 , 2 , . . . , n } Des π : { i ∈ { 1 , . . . n − 1 } | π ( i ) > π ( i + 1 ) } ( descents ) des π : | Des π | , the number of descents. π ∈ S 3 Des π des π 1 2 3 { } 0 1 3 2 { 2 } 1 2 1 3 { 1 } 1 2 3 1 { 2 } 1 3 1 2 { 1 } 1 3 2 1 { 1 , 2 } 2

Permutations and Descents S n : set of permutations π : { 1 , 2 , . . . , n } → { 1 , 2 , . . . , n } Des π : { i ∈ { 1 , . . . n − 1 } | π ( i ) > π ( i + 1 ) } ( descents ) des π : | Des π | , the number of descents. Descent polynomial: π ∈ S 3 Des π des π 1 2 3 { } 0 � x des π E n ( x ) = 1 3 2 { 2 } 1 π ∈ S n 2 1 3 { 1 } 1 2 3 1 { 2 } 1 E 3 ( x ) = 1 + 4 x + x 2 3 1 2 { 1 } 1 3 2 1 { 1 , 2 } 2

Permutations and Descents S n : set of permutations π : { 1 , 2 , . . . , n } → { 1 , 2 , . . . , n } Des π : { i ∈ { 1 , . . . n − 1 } | π ( i ) > π ( i + 1 ) } ( descents ) des π : | Des π | , the number of descents. Descent polynomial: π ∈ S 3 Des π des π 1 2 3 { } 0 � x des π E n ( x ) = 1 3 2 { 2 } 1 π ∈ S n 2 1 3 { 1 } 1 2 3 1 { 2 } 1 E 3 ( x ) = 1 + 4 x + x 2 3 1 2 { 1 } 1 3 2 1 { 1 , 2 } 2 Eulerian polynomials: E n ( x )

The Eulerian polynomials, E n ( x ) � x des π E n ( x ) = π ∈ S n E n ( x ) � ( t + 1 ) n x t = ( 1 − x ) n + 1 t ≥ 0 E n ( x ) z n ( 1 − x ) � = e z ( x − 1 ) − x n ! n ≥ 0

Inversion Sequences I n = { ( e 1 , . . . e n ) ∈ Z n | 0 ≤ e i < i } Encode permutations as inversion sequences φ : S n → I n φ ( π ) = ( e 1 , . . . , e n ) , where e j = |{ i | i < j and π ( i ) > π ( j ) }| . Example: φ ( 4 3 6 5 1 2 ) = ( 0 , 1 , 0 , 1 , 4 , 4 ) .

Inversion Sequences I n = { ( e 1 , . . . e n ) ∈ Z n | 0 ≤ e i < i } Encode permutations as inversion sequences φ : S n → I n φ ( π ) = ( e 1 , . . . , e n ) , where e j = |{ i | i < j and π ( i ) > π ( j ) }| . Example: φ ( 4 3 6 5 1 2 ) = ( 0 , 1 , 0 , 1 , 4 , 4 ) . Claim: φ is a bijection with Des π = Asc φ ( π ) .

Inversion Sequences I n = { ( e 1 , . . . e n ) ∈ Z n | 0 ≤ e i < i } Encode permutations as inversion sequences φ : S n → I n φ ( π ) = ( e 1 , . . . , e n ) , where e j = |{ i | i < j and π ( i ) > π ( j ) }| . Example: φ ( 4 3 6 5 1 2 ) = ( 0 , 1 , 0 , 1 , 4 , 4 ) . Claim: φ is a bijection with Des π = Asc φ ( π ) . What is “ Asc ”?

Ascents What is an “ascent” in an inversion sequence? e i < e i + 1 ?

Ascents What is an “ascent” in an inversion sequence? e i i < e i + 1 e i < e i + 1 ? or i + 1 ?

Ascents What is an “ascent” in an inversion sequence? e i i < e i + 1 e i < e i + 1 ? or i + 1 ? Lemma If 0 ≤ e j < j for all j ≤ n, then for 1 ≤ i < n, e i i < e i + 1 e i < e i + 1 iff i + 1 .

View inversion sequences as lattice points in a (half-open) 1 × 2 × · · · × n box View ascent constraints as hyperplane constraints: 0 < e 1 and e i i < e i + 1 i + 1 , 1 ≤ i < n π ∈ S 3 e ∈ I n Asc e 1 2 3 (0,0,0) { } 1 3 2 (0,0,1) { 2 } 2 1 3 (0,1,0) { 1 } 2 3 1 (0,0,2) { 2 } 3 1 2 (0,1,1) { 1 } 3 2 1 (0,1,2) { 1 , 2 }

s-inversion sequences For any sequence s = ( s 1 , s 2 , . . . , s n ) of positive integers: { ( e 1 , . . . , e n ) ∈ Z n | 0 ≤ e i < s i for 1 ≤ i ≤ n } . I ( s ) = n (lattice points in a half-open s 1 × s 2 × · · · × s n box) � � � I ( s ) � = s 1 s 2 · · · s n � � n

s-inversion sequences For any sequence s = ( s 1 , s 2 , . . . , s n ) of positive integers: { ( e 1 , . . . , e n ) ∈ Z n | 0 ≤ e i < s i for 1 ≤ i ≤ n } . I ( s ) = n An ascent of e is a position i : 1 ≤ i < n and e i e i + 1 < . s i s i + 1 If e 1 > 0 then 0 is an ascent . %pause

s-inversion sequences For any sequence s = ( s 1 , s 2 , . . . , s n ) of positive integers: { ( e 1 , . . . , e n ) ∈ Z n | 0 ≤ e i < s i for 1 ≤ i ≤ n } . I ( s ) = n Example: ( 2 , 4 , 5 ) ∈ I ( 3 , 5 , 7 ) n Asc e = { 0 , 1 } 2 �∈ Asc e since 4 / 5 � < 5 / 7 %pause

Ascent polynomials of s -inversion sequences A ( s ) � x asc e n ( x ) = e ∈ I ( s ) n ( 2 , 4 ) -inversion sequences Ascent sets: { } yellow dot { 0 } blue square { 1 } red diamond { 0 , 1 } black dot A ( 2 , 4 ) ( x ) = 1 + 6 x + x 2 n

Ascent polynomials of s -inversion sequences A ( s ) � x asc e n ( x ) = e ∈ I ( s ) n ( 3 , 5 ) -inversion sequences ( 2 , 4 ) -inversion sequences Ascent sets: { } yellow dot { 0 } blue square { 1 } red diamond { 0 , 1 } black dot A ( 3 , 5 ) ( x ) = 1 + 10 x + 4 x 2 A ( 2 , 4 ) ( x ) = 1 + 6 x + x 2 n n

Call A ( s ) n ( x ) the s-Eulerian polynomial since, when s = ( 1 , 2 , . . . , n ) , x asc e = x des π = E n ( x ) , A ( s ) � � n ( x ) = e ∈ I ( s ) π ∈ S n n the Eulerian polynomial. (Recall bijection φ : S n → I n with Des π = Asc φ ( π ) )

Why s -inversion sequences? • Natural model for combinatorial structures • Can prove general properties of the s -Eulerian polynomials • Surprising results follow using Ehrhart theory • Can be encoded as lecture hall partitions • Lead to a natural refinement of the s -Eulerian polynomials • Help answer questions about lecture hall partitions

sequence s s -Eulerian polynomial 1 + 57 x + 302 x 2 + 302 x 3 + 57 x 4 + x 5 ( 1 , 2 , 3 , 4 , 5 , 6 ) 1 + 237 x + 1682 x 2 + 1682 x 3 + 237 x 4 + x 5 ( 2 , 4 , 6 , 8 , 10 ) 1 + 57 x + 302 x 2 + 302 x 3 + 57 x 4 + x 5 ( 6 , 5 , 4 , 3 , 2 , 1 ) 1 + 20 x + 48 x 2 + 20 x 3 + x 4 ( 1 , 1 , 3 , 2 , 5 , 3 ) 1 + 358 x + 3580 x 2 + 5168 x 3 + 1328 x 4 + 32 x 5 . ( 1 , 3 , 5 , 7 , 9 , 11 ) 1 + 71 x + 948 x 2 + 2450 x 3 + 1411 x 4 + 159 x 5 ( 7 , 2 , 3 , 5 , 4 , 6 )

1. Signed permutations and ( 2 , 4 , 6 , . . . , 2 n ) -inversion sequences � � � I ( 2 , 4 , 6 ,..., 2 n ) � = 2 n n ! � � n

Signed Permutations B n B n = { ( σ 1 , . . . , σ n ) | ∃ π ∈ S n , ∀ i σ i = ± π ( i ) } π ∈ B n Des π ( 1, 2) { } (-1, 2) { 0 } ( 1,-2) { 1 } (-1,-2) { 0 , 1 } ( 2, 1) { 1 } Des σ = { i ∈ { 0 , . . . , n − 1 } | σ i > σ i + 1 } , (-2, 1) { 0 } ( 2,-1) { 1 } (-2,-1) { 0 } with the convention that σ 0 = 0. descent polynomial: 1 + 6 x + x 2

Signed Permutations B n B n = { ( σ 1 , . . . , σ n ) | ∃ π ∈ S n , ∀ i σ i = ± π ( i ) } ( 2 , 4 ) -inversion sequences π ∈ B n Des π ( 1, 2) { } (-1, 2) { 0 } ( 1,-2) { 1 } (-1,-2) { 0 , 1 } Ascent sets: ( 2, 1) { 1 } { } yellow dot (-2, 1) { 0 } { 0 } blue square ( 2,-1) { 1 } { 1 } red diamond (-2,-1) { 0 } { 0 , 1 } black dot descent polynomial: ascent polynomial: 1 + 6 x + x 2 1 + 6 x + x 2

Theorem (Pensyl, S 2012/13) x des σ = A ( 2 , 4 ,..., 2 n ) � ( x ) . n σ ∈ B n

Theorem (Pensyl, S 2012/13) x des σ = A ( 2 , 4 ,..., 2 n ) � ( x ) . n σ ∈ B n Proof. There is a bijection Θ : B n → I ( 2 , 4 ,..., 2 n ) satisfying n Des σ = Asc Θ( σ ) .

2. ( s 1 , s 2 , . . . , s n ) -inversion sequences vs. ( s n , s n − 1 , . . . , s 1 ) -inversion sequences Example from table: 1 + 57 x + 302 x 2 + 302 x 3 + 57 x 4 + x 5 A ( 1 , 2 , 3 , 4 , 5 , 6 ) ( x ) = 6 A ( 6 , 5 , 4 , 3 , 2 , 1 ) = ( x ) 6

Theorem (S, Schuster 2012; Liu, Stanley 2012) For any sequence ( s 1 , s 2 , . . . , s n ) of positive integers, A ( s 1 , s 2 ,..., s n ) ( x ) = A ( s n , s n − 1 ,..., s 1 ) ( x ) . n n

Reversing s preserves the ascent polynomial ( 3 , 5 ) -inversion sequences ( 5 , 3 ) -inversion sequences Ascent sets: { } yellow dot { 0 } blue square { 1 } red diamond { 0 , 1 } black dot 1 + 10 x + 4 x 2 1 + 10 x + 4 x 2 but not necessarily the partition into ascent sets

3. Roots of s -Eulerian polynomials Example from table: ( x ) = 1 + 71 x + 948 x 2 + 2450 x 3 + 1411 x 4 + 159 x 5 A ( 7 , 2 , 3 , 5 , 4 , 6 ) 6 Roots in the intervals: [[ − 19 / 1024 , − 9 / 512 ] , [ − 77 / 1024 , − 19 / 256 ] , [ − 423 / 1024 , − 211 / 512 ] , [ − 1701 / 1024 , − 425 / 256 ] , [ − 3435 / 512 , − 6869 / 1024 ]]

Theorem (S, Visontai 2012) For every sequence s of positive integers, A ( s ) n ( x ) has all real roots.

Theorem (S, Visontai 2012) For every sequence s of positive integers, A ( s ) n ( x ) has all real roots. Corollary (Frobenius 1910; Brenti 1994) The descent polynomials for Coxeter groups of types A and B have all real roots.

Theorem (S, Visontai 2012) For every sequence s of positive integers, A ( s ) n ( x ) has all real roots. Corollary (Frobenius 1910; Brenti 1994) The descent polynomials for Coxeter groups of types A and B have all real roots. New: ([S, Visontai 2013]) Method can be adapted to type D .

Theorem (S, Visontai 2012) For every sequence s of positive integers, A ( s ) n ( x ) has all real roots.

Theorem (S, Visontai 2012) For every sequence s of positive integers, A ( s ) n ( x ) has all real roots. Corollary For any s, the sequence of coefficents of the s-Eulerian polynomial is unimodal and log-concave. Example A ( 7 , 2 , 3 , 5 , 4 , 6 ) ( x ) : 6 1 , 71 , 948 , 2450 , 1411 , 159 , 1

4. Lecture Hall Polytopes and Ehrhart Theory