BARIL & DO ECO-generation for compositions and Introduction their restrictions Recalls Bijection C 1 , p ( n ) Jean-Luc BARIL & Phan-Thuan DO p ( n ) C ˆ C # p ( n ) Université de Bourgogne – France Summary CAT Permutation Patterns 2008 Dunedin, 16-20 Juin 1 / 26
Introduction BARIL & DO Introduction Recalls A composition c of n can be written as c = ( c 1 , c 2 , . . . , c k ) with Bijection c 1 + c 2 + . . . + c k = n and c i ≥ 1 , ∀ i ≤ k . C 1 , p ( n ) p ( n ) C ˆ C # p ( n ) Summary CAT 2 / 26
Introduction BARIL & DO Introduction Recalls A composition c of n can be written as c = ( c 1 , c 2 , . . . , c k ) with Bijection c 1 + c 2 + . . . + c k = n and c i ≥ 1 , ∀ i ≤ k . C 1 , p ( n ) C ( n ) is the set of compositions of an integer n p ( n ) C ˆ C ≤ p ( n ) is the set of compositions of n with all parts of sizes ≤ p is the set of ( 1 , p ) -compositions of n C 1 , p ( n ) C # p ( n ) p ( n ) is the set of compositions of n without parts of size p C ˆ C # p ( n ) is the set of compositions of n with at most p parts C ∗ ( n , p , r ) is the set of compositions of n with the last part of size = r mod p Summary C ( n , p , r ) is the set of compositions of n with all parts of size = r mod p CAT 2 / 26
Some bibliographies BARIL & DO Alladi and Hoggatt, Compositions with ones and twos , 1975. Introduction Carlitz, Restricted compositions , 1976. Chinn and Heubach, ( 1 , k ) -Compositions , 2003. Recalls Chinn and Heubach, Compositions of n with no occurrence of k , 2003. Bijection . . . C 1 , p ( n ) p ( n ) C ˆ C # p ( n ) Summary CAT 3 / 26
Some bibliographies BARIL & DO Alladi and Hoggatt, Compositions with ones and twos , 1975. Introduction Carlitz, Restricted compositions , 1976. Chinn and Heubach, ( 1 , k ) -Compositions , 2003. Recalls Chinn and Heubach, Compositions of n with no occurrence of k , 2003. Bijection . . . C 1 , p ( n ) p ( n ) C ˆ Klingsberg, A Gray code for compositions , 1982. Walsh, Loop-free sequencing of bounded integer compositions , 2000. C # p ( n ) Vajnovszki, A loopless generation of bitstrings without p consecutive ones , 2001. Summary Baril and Moreira, More restrictive Gray code for (1,p)-compositions and CAT relatives , 2008. 3 / 26
Some bibliographies BARIL & DO Alladi and Hoggatt, Compositions with ones and twos , 1975. Introduction Carlitz, Restricted compositions , 1976. Chinn and Heubach, ( 1 , k ) -Compositions , 2003. Recalls Chinn and Heubach, Compositions of n with no occurrence of k , 2003. Bijection . . . C 1 , p ( n ) p ( n ) C ˆ Klingsberg, A Gray code for compositions , 1982. Walsh, Loop-free sequencing of bounded integer compositions , 2000. C # p ( n ) Vajnovszki, A loopless generation of bitstrings without p consecutive ones , 2001. Summary Baril and Moreira, More restrictive Gray code for (1,p)-compositions and CAT relatives , 2008. Barcucci et al., ECO : a methodology for the Enumeration of Combinatorial Objects , 1999. 3 / 26
Recalls BARIL & DO Introduction Recalls Bijection ECO method - Generating tree C 1 , p ( n ) Pattern avoiding permutations p ( n ) C ˆ Active sites - Right justified sites Regular class - c -Regular class C # p ( n ) Succession functions - General generating algorithm Summary CAT 4 / 26
Enumerating Combinatorial Object Methods Barcucci, Del Lungo, Pergola, Pinzani 1999 BARIL & DO Introduction The ECO method is used for the enumeration and the Recalls recursive construction of combinatorial object classes. Bijection C 1 , p ( n ) This is a recursive description of a combinatorial object class which explains how an object of size n can be p ( n ) C ˆ reached from one and only one object of inferior size. C # p ( n ) Summary CAT 5 / 26
Enumerating Combinatorial Object Methods Barcucci, Del Lungo, Pergola, Pinzani 1999 BARIL & DO Introduction The ECO method is used for the enumeration and the Recalls recursive construction of combinatorial object classes. Bijection C 1 , p ( n ) This is a recursive description of a combinatorial object class which explains how an object of size n can be p ( n ) C ˆ reached from one and only one object of inferior size. C # p ( n ) It consists to give a system of succession rules for a combinatorial object class which induces a generating Summary tree such that each node is labeled : the set of CAT successions rules describes for each node the label of its successors. 5 / 26
Pattern BARIL & DO Introduction Recalls S n - the set of permutations on [ n ] = { 1 , 2 , . . . , n } . Bijection Let a = a 1 . . . a k . The pattern of a is the permutation C 1 , p ( n ) τ ∈ S k obtained from a by substituting the minimum p ( n ) element by 1, the second minimum element by 2, . . . , C ˆ and the maximum element by k . C # p ( n ) Summary Example CAT The pattern of a = 914 is τ = 312 . 6 / 26
Pattern Avoiding Permutation For a τ ∈ S k and a π ∈ S n , π is τ -avoiding iff there is no BARIL & DO subsequence π ( i 1 ) π ( i 1 ) . . . π ( i k )( i 1 < i 2 < . . . < i k ) whose Introduction pattern is τ . We write S n ( τ ) for the set of τ -avoiding Recalls permutations of [ n ] . Bijection C 1 , p ( n ) Example π = 512634 avoids 321-pattern. p ( n ) C ˆ But π contains 3412-pattern ( 512634 ). C # p ( n ) Summary A barred permutation pattern is a permutation pattern in CAT which overbars are used to indicate that barred values cannot occur at the barred positions. Example π = 5716342 fails to be 4132-avoiding but is 4132 -avoiding. 7 / 26
Active Sites The sites of π ∈ S n ( T ) are the positions between two BARIL & DO consecutive elements, before the first and after the last Introduction element. Recalls The sites are numbered, from right to left, from 1 to n + 1 . Bijection i is an active site of π ∈ S n ( T ) if the permutation C 1 , p ( n ) obtained from π by inserting n + 1 into its i th site is a permutation in S n + 1 ( T ) . p ( n ) C ˆ C # p ( n ) Summary CAT 8 / 26
Active Sites The sites of π ∈ S n ( T ) are the positions between two BARIL & DO consecutive elements, before the first and after the last Introduction element. Recalls The sites are numbered, from right to left, from 1 to n + 1 . Bijection i is an active site of π ∈ S n ( T ) if the permutation C 1 , p ( n ) obtained from π by inserting n + 1 into its i th site is a permutation in S n + 1 ( T ) . p ( n ) C ˆ χ T ( i , π ) - the number of active sites of the permutation C # p ( n ) obtained from π by inserting n + 1 into its i th active site. Summary CAT 8 / 26
Active Sites The sites of π ∈ S n ( T ) are the positions between two BARIL & DO consecutive elements, before the first and after the last Introduction element. Recalls The sites are numbered, from right to left, from 1 to n + 1 . Bijection i is an active site of π ∈ S n ( T ) if the permutation C 1 , p ( n ) obtained from π by inserting n + 1 into its i th site is a permutation in S n + 1 ( T ) . p ( n ) C ˆ χ T ( i , π ) - the number of active sites of the permutation C # p ( n ) obtained from π by inserting n + 1 into its i th active site. The active sites of a permutation π ∈ S n ( T ) are right Summary justified if the sites to the right of any active site are also CAT active. Example 13452 ∈ S 5 ( 312 ) has 3 first active sites right justified following 134 _ 5 _ 2 _. 8 / 26
Regular pattern A set of patterns T is called regular if BARIL & DO 1 ∈ S 1 ( T ) has two sons, Introduction Recalls all active sites are right justified, Bijection for any n ≥ 1 and π ∈ S n ( T ) , χ T ( i , π ) does not depend on π C 1 , p ( n ) but solely on i and on the number k of active sites of π . In this case we denote χ T ( i , π ) by χ T ( i , k ) and we call it succession p ( n ) C ˆ function [Do, Vajnovszki 2007] . C # p ( n ) ( k ) � ( χ T ( 1 , k ))( χ T ( 2 , k )) . . . ( χ T ( k , k )) Summary or ( k ) � ∪ k i = 1 ( χ T ( i , k )) , for k ≥ 1 , CAT is the succession rule corresponding to the set of patterns T . succession function → succession rule succession rule � succession function 9 / 26
1 12 21 123 132 213 231 321 Example 1234 1243 1324 1342 1432 2134 2143 2314 2341 2431 3214 3241 3421 4321 BARIL & DO The Catalan sets of permutations avoiding pattern T = { 312 } and T = { 321 } have the same succession rule Introduction Recalls ( k ) � ( 2 )( 3 ) . . . ( k + 1 ) , but different succession functions : Bijection T = { 312 } , χ T ( i , k ) = i + 1 C 1 , p ( n ) p ( n ) C ˆ C # p ( n ) Summary CAT 10 / 26
1 12 21 123 132 312 213 231 1234 1243 1423 4123 1324 1342 3124 3142 3412 2134 2143 2413 2314 2341 BARIL & DO � k + 1 Introduction if i = 1 T = { 321 } , χ T ( i , k ) = Recalls i otherwise Bijection C 1 , p ( n ) p ( n ) C ˆ C # p ( n ) Summary CAT 11 / 26
Colored regular pattern BARIL & DO Introduction Recalls color : each permutation associated with an integer c Bijection [Barcucci, Pinzani, . . . ] . C 1 , p ( n ) if π ∈ S n ( T ) with color c , the insertion of n + 1 in its i -th p ( n ) C ˆ active site produces a σ ∈ S n + 1 ( T ) with µ ( i , π, c ) active sites and color ν ( i , π, c ) ; C # p ( n ) we extend the previous χ function in order to transform a Summary triple ( i , k , c ) ∈ N 3 into a couple ( µ ( i , k , c ) , ν ( i , k , c )) ∈ N 2 . CAT 12 / 26
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