A sextuple equidistribution arising in Pattern Avoidance Zhicong - PowerPoint PPT Presentation

A sextuple equidistribution arising in Pattern Avoidance Zhicong Lin NIMS & Jimei University 78th S´ eminaire Lotharingien de Combinatoire March 29, 2017 Joint work with Dongsu Kim Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Eulerian polynomials Definition The Eulerian polynomial A n ( t ) may be defined by Euler’s basic formula (Leonhard Euler 1755): A n ( t ) ( k + 1) n t k = � (1 − t ) n +1 . k ≥ 0 A 1 ( t ) = 1 A 2 ( t ) = 1 + t A 3 ( t ) = 1 + 4 t + t 2 A 4 ( t ) = 1 + 11 t + 11 t 2 + t 3 A 5 ( t ) = 1 + 26 t + 66 t 2 + 26 t 3 + t 4 Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Permutation Statistics S n : Set of permutations of [ n ] := { 1 , 2 , · · · , n } Definition For π = π 1 π 2 · · · π n ∈ S n : DES ( π ) := { i ∈ [ n − 1] : π i > π i +1 } des ( π ) := | DES ( π ) | (Descent number) . DES (3 . 15 . 24) = { 1 , 3 } Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Permutation Statistics S n : Set of permutations of [ n ] := { 1 , 2 , · · · , n } Definition For π = π 1 π 2 · · · π n ∈ S n : DES ( π ) := { i ∈ [ n − 1] : π i > π i +1 } des ( π ) := | DES ( π ) | (Descent number) . DES (3 . 15 . 24) = { 1 , 3 } Theorem (Riordan 1958) � t des ( π ) . A n ( t ) = π ∈ S n Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Inversion sequences Inversion sequences: I n = { ( e 1 , e 2 , . . . , e n ) ∈ Z n : 0 ≤ e i < i } I 3 = { (0 , 0 , 0) , (0 , 0 , 1) , (0 , 0 , 2) , (0 , 1 , 0) , (0 , 1 , 1) , (0 , 1 , 2) } Definition For e = ( e 1 , e 2 , · · · , e n ) ∈ I n : ASC ( e ) := { i ∈ [ n − 1] : e i < e i +1 } asc( e ) := | ASC ( e ) | (Ascent number) . ASC (0 , 1 , 1 , 2 , 0) = { 1 , 3 } Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

A natural bijection: inv -code | S n | = | I n | = n ! and more... t des ( π ) = � � t asc( e ) π ∈ S n e ∈ I n Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

A natural bijection: inv -code | S n | = | I n | = n ! and more... t des ( π ) = � � t asc( e ) π ∈ S n e ∈ I n A natural bijection ( inv -code) φ : S n → I n with φ ( π ) = ( e 1 , . . . , e n ), where e i = |{ j : j < i and π j > π i }| . Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

A natural bijection: inv -code | S n | = | I n | = n ! and more... t des ( π ) = � � t asc( e ) π ∈ S n e ∈ I n A natural bijection ( inv -code) φ : S n → I n with φ ( π ) = ( e 1 , . . . , e n ), where e i = |{ j : j < i and π j > π i }| . This proves even more: t DES ( π ) = � � t ASC( e ) , π ∈ S n e ∈ I n where t { i 1 ,..., i k } := t i 1 · · · t i k . Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Double Eulerian statistics dist( e ): number of distinct positive entries in e Theorem (Dumont 1974) t des ( π ) = � � t dist( e ) . π ∈ S n e ∈ I n Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Double Eulerian statistics dist( e ): number of distinct positive entries in e Theorem (Dumont 1974) t des ( π ) = � � t dist( e ) . π ∈ S n e ∈ I n Via V-code and S-code: Theorem (Foata 1977) s des ( π − 1 ) t DES ( π ) = � � s dist( e ) t ASC( e ) . π ∈ S n e ∈ I n Rediscovered by Visontai (2013) An essentially different proof by Aas in PP 2013 (Paris) Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Gessel’s γ -positivity conjecture Double Eulerian polynomials (Carlitz-Roselle-Scoville 1966): s des ( π − 1 ) t des ( π ) . � A n ( s , t ) := π ∈ S n Conjectured by Gessel (2005): Theorem (L. 2015) The integers γ n , i , j are nonnegative in: � γ n , i , j ( st ) i (1 + st ) j ( s + t ) n − 1 − j − 2 i . A n ( s , t ) = i , j ≥ 0 j +2 i ≤ n − 1 Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Permutations without double descents 8 r � ❅ r 7 � ✛ � ❅ ❅ r 6 � � ✲ 5 ❅ ❅ � � ✲ r 4 ❅ ❅ � � r ✲ 3 ❅ ❅ � r ❅ r 2 � ❅ � ✛ r 1 ❅ � ❅ −∞ −∞ Figure : Foata-Strehl actions on 34862571 NDD n : set of all permutations in S n without double descents Theorem (Foata & Sch¨ utzenberger 1970) ⌊ ( n − 1) / 2 ⌋ � γ n , i t i (1 + t ) n +1 − 2 i , A n ( t ) = i =0 where γ n , i = # { π ∈ NDD n : des ( π ) = i } . Problem Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Permutations without double descents NDD n : set of all permutations in S n without double descents Theorem (Foata & Sch¨ utzenberger 1970) ⌊ ( n − 1) / 2 ⌋ � γ n , i t i (1 + t ) n +1 − 2 i , A n ( t ) = i =0 where γ n , i = # { π ∈ NDD n : des ( π ) = i } . Problem Is there any combinatorial interpretation for γ n , i , j ? Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Separable permutations Restrict to the terms without s + t : π = 2413 ❍❍ r des ( π ) = 1 ❇ ❇ ❍ des ( π − 1 ) = 2 ❇ r ❇ ❍❍ First des ( π ) � = des ( π − 1 ) r ❍ ❇ r Definition Permutations that avoid both the patterns 2413 and 3142 are separable permutations. West (1995): | S n (2413 , 3142) | = S n , the n th Large Schr¨ oder numbers. Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Separable permutations Separable permutations “di-sk” trees ⊖ ⊖ ⊕ bij. ⇐ ⇒ ⊖ ⊖ ⊕ ⊕ ⊖ Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Descent polynomial on Separable permutations Via combinatorial approach using “di-sk” trees: Theorem (Fu-L.-Zeng 2015) ⌊ ( n − 1) / 2 ⌋ t des ( π ) = � � γ S n , k t k (1 + t ) n − 1 − 2 k , π ∈ S n (2413 , 3142) k =0 where γ S n , k = |{ π ∈ S n (3142 , 2413) ∩ NDD n : des ( π ) = k }| . Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

021-avoiding inversion sequences 021-avoiding ⇔ positive entries are weakly increasing Via bijections with “di-sk” trees: Theorem (Fu-L.-Zeng & Corteel et al. 2015) t des ( π ) = � � t asc( e ) . π ∈ S n (2413 , 3142) e ∈ I n (021) Problem ⌊ ( n − 1) / 2 ⌋ t asc( e ) = � � γ S n , k t k (1 + t ) n − 1 − 2 k k =0 e ∈ I n (021) What is the combinatorial interpretation of γ S n , k in terms of 021 -avoiding inversion sequences? Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Double Eulerian equidistribution Theorem (Foata 1977) s des ( π − 1 ) t DES ( π ) = � � s dist( e ) t ASC( e ) . π ∈ S n e ∈ I n Restricted version of Foata’s 1977 result: Theorem (Kim-L. 2016) s des ( π − 1 ) t DES ( π ) = � � s dist( e ) t ASC( e ) . π ∈ S n (2413 , 4213) e ∈ I n (021) Neither Foata’s original bijection nor Aas’ approach could be applied to prove this restricted version. Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

First application 8 r � ❅ r 7 � ✛ � ❅ ❅ r 6 � � ✲ ❅ ❅ 5 � � r ✲ 4 ❅ ❅ � r � ✲ 3 ❅ ❅ � r ❅ r 2 � ❅ � ✛ r 1 ❅ � ❅ −∞ −∞ As S n (2413 , 4213) is invariant under Foata-Strehl action: Corollary ⌊ ( n − 1) / 2 ⌋ t asc( e ) = � � γ S n , k t k (1 + t ) n − 1 − 2 k , e ∈ I n (021) k =0 where γ S n , k = |{ e ∈ I n (021) : e has no double ascents , asc( e ) = k }| . Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Second application d �→ 0 1 0 1 2 0 4 Figure : The outline of an inversion sequence s des ( π − 1 ) t des ( π ) � z n � S = S ( s , t ; z ) := n ≥ 1 π ∈ S n (2413 , 4213) Theorem (Double Eulerian distribution) S = t ( z ( s − 1) + 1) S + tz (2 s − 1) S 2 + z ( ts + 1) S + z . Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

Ascents on Schr¨ oder paths A Schr¨ oder n -path is a lattice path on the plane from (0 , 0) to (2 n , 0), never going below x -axis, using the steps (1 , 1) (1 , − 1) (2 , 0) . ✉ � ❅ � ❅ � ❅ ✉ ✉ ✉ ✉ � ❅ � ❅ � � ❅ ❅ � ❅ ✉ � ❅ ✉ ✉ ✉ Corollary (Conjecture of Corteel et al. 2015) An ascent in a Schr¨ oder path is a maximal string of consecutive up steps. Denoted by SP n the set of Schr¨ oder n-path and by asc( p ) the number of ascents of p. Then, s dist( e ) = � � s asc( p ) . p ∈ SP n − 1 e ∈ I n (021) Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance

A sextuple equidistribution (Statistics) For each π ∈ S n : VID ( π ) := { 2 ≤ i ≤ n : π i appears to the right of ( π i + 1) } , the v alues of i nverse d escents of π ; LMA ( π ) := { i ∈ [ n ] : π i > π j for all 1 ≤ j < i } , the positions of l eft-to-right ma xima of π ; LMI ( π ) := { i ∈ [ n ] : π i < π j for all 1 ≤ j < i } , the positions of l eft-to-right mi nima of π ; RMA ( π ) := { i ∈ [ n ] : π i > π j for all j ≥ i } , the positions of r ight-to-left ma xima of π ; RMI ( π ) := { i ∈ [ n ] : π i < π j for all j ≥ i } , the positions of r ight-to-left mi nima of π ; Zhicong Lin A sextuple equidistribution arising in Pattern Avoidance