Rational Catalan Numbers and Music Franck Rational Catalan Numbers and Music Jedrzejewski Catalan Numbers Franck Jedrzejewski Rational Catalan Numbers Dyck Path Paris-Saclay University - CEA - France Christoffel words Well-formed Scales IRMA Strasbourg, Mars 29, 2019 Narayana Numbers Block Designs Johson Works Catalan Designs Permutations Rational Associahedra 1 / 47
Rational Contents Catalan Numbers and Music Franck Jedrzejewski Catalan Numbers Rational Catalan Numbers 1 Catalan Numbers Dyck Path 2 Rational Catalan Numbers Christoffel words 3 Dyck Paths Well-formed Scales 4 Well-Formed Scales Narayana 5 Noncrossing Partitions Numbers 6 Associahedra Block Designs 7 Parking Functions Johson Works Catalan Designs 8 Combinatorial t-designs Permutations 9 Catalan Designs Rational 10 Rational Associahedra Associahedra Charles Eugène Catalan (1814-1894) 2 / 47
Rational Catalan Numbers Catalan Numbers and Music � 2 n n Franck 1 (2 n )! n + k � Jedrzejewski � C n = = ( n + 1)! n ! = n + 1 n k Catalan k =2 Numbers The first Catalan numbers for n = 0, 1, 2, 3, ... are : Rational Catalan 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, Numbers etc. Dyck Path Christoffel words Recurrence relations : Well-formed Scales n � C 0 = 1 , C n +1 = C k C n − k Narayana Numbers k =0 Block Designs C n +1 = 2(2 n + 1) Johson Works C 0 = 1 , C n n + 2 Catalan Designs Asymptotic behavior : Permutations Rational 4 n Associahedra C n ∼ n 3 / 2 √ π Integral representation : � 4 � ρ ( x ) = 1 4 − x x n ρ ( x ) dx , C n = 2 π x 0 3 / 47
Rational Catalan Family Catalan Numbers and Music Franck Jedrzejewski Catalan Numbers Rational Catalan Numbers Dyck Path Christoffel words Well-formed Scales Narayana Numbers Block Designs Johson Works Catalan Designs Permutations Rational Associahedra 4 / 47
Rational Grafting Catalan Numbers and Music Franck Jedrzejewski Associahedra = representation of the algebra of planar rooted binary trees = Catalan dendriform algebra (Jean-Louis Loday) Numbers Rational Catalan Numbers Dyck Path Christoffel words Well-formed Scales Narayana Numbers Block Designs Johson Works Catalan Designs Permutations Rational Associahedra 5 / 47
Rational Rational Catalan Numbers Catalan Numbers and Music Franck Given x ∈ Q \ [ − 1 , 0] , there exist a unique coprime ( a , b ) ∈ N 2 Jedrzejewski such that a Catalan x = Numbers b − a Rational Catalan The Rational Catalan Number : Numbers Dyck Path � a + b Christoffel words 1 � = ( a + b − 1)! Cat ( x ) = Cat ( a , b ) = Well-formed a + b a , b a ! b ! Scales Nikolaus von Fuss Narayana (1755-1826) Numbers Block Designs Special Cases : Johson Works Catalan Designs 1 a = n , b = n + 1 Eugène Charles Catalan (1814-1894) Permutations Rational (2 n )! Associahedra Cat ( n ) = Cat ( n , n + 1) = ( n + 1)! n ! = C n 2 a = n , b = kn + 1 Nikolaus von Fuss (1755-1826) � ( k + 1) n + 1 Cat ( a , b ) = (( k + 1) n )! 1 � ( kn + 1)! n ! = ( k + 1) n + 1 n 6 / 47
Rational Derived Catalan number Catalan Numbers and Music The commutativity Cat ( a , b ) = Cat ( b , a ) = ( a + b − 1)! implies that the derived a ! b ! Franck Catalan Number satisfies : Jedrzejewski � 1 � � x � Cat ′ ( x ) := Cat = Cat Catalan x − 1 1 − x Numbers Rational Duality : Rational Catalan Numbers Cat ′ � 1 Dyck Path 1 x � � � � � = Cat ′ ( x ) Christoffel words = Cat = Cat = x 1 / x − 1 1 − x Well-formed Scales The process Cat ( x ) → Cat ′ ( x ) → Cat ′′ ( x ) ... is a categorification of the Euclidean Narayana Numbers algorithm Block Designs Euclidean Algorithm : Johson Works b = aq 0 + r 0 , a = q 1 r 0 + r 1 , r 0 = q 2 r 1 + r 2 , ..., r n = q n +2 r n +1 + r n +2 Catalan Designs Permutations g = gcd( b , a ) = gcd( a , r 0 ) = gcd( r 0 , r 1 ) = · · · = gcd( r n , r n +1 ) = r n +2 Rational Associahedra Catalan Algorithm : for the minor third x = 6 / 5 , ( a , b ) = (5 , 11) Cat (5 , 11) = 143 Cat ′ (5 , 11) = Cat (5 , 6) = 42 Cat ′′ (5 , 11) Cat ′ (5 , 6) = Cat (1 , 5) = 1 = 7 / 47
Rational Dyck Words and Dyck Paths Catalan Numbers and Music Franck Jedrzejewski Dyck words ( = Well parenthesized words) Catalan Numbers alphabet Σ = { ( , ) } , imb ( ω ) = | ω | ( − | ω | ) Rational Catalan Numbers Dyck Path ω is a Dyck word iff imb ( ω ) = 0 and imb ( u ) ≥ 0 for all Christoffel words prefix u of ω Well-formed Scales Dyck path from (0,0) to ( a , b ) = staircase walk that Narayana lies below the diagonal (but may touch). Numbers Walther von Dyck Block Designs (1856-1934) Johson Works Catalan Designs Theorem (Grossman (1950), Bizley (1954)) Permutations The number of Dyck paths is the Catalan number : Rational Associahedra | D ( x ) | = Cat ( x ) H. D. Grossman. Fun with lattice points : paths in a lattice triangle, Scripta Math. 16 (1950) 207–212 M. T. L. Bizley. Derivation of a new formula for the number of minimal lattice paths from (0,0) to ( km , kn ) having just tcontacts with the line my = nx and having no points above this line ; and a proof of Grossmans formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries . 80 (1954) 55–62 8 / 47
Rational Rational Dyck Paths Catalan Numbers and Music Franck Jedrzejewski Catalan Numbers Rational Catalan Numbers Dyck Path Christoffel words Well-formed Scales Narayana Numbers Block Designs Dyck Paths = Path from (0,0) to ( b , a ) in the integer lattice Z 2 staying above the Johson Works diagonal y = ax / b . Catalan Designs Permutations Bottom of a north step (blue) by laser construction gives the dissection of P b +1 Rational Associahedra Dyck Path in red : xyxy 2 xy 2 Number of ( a , b )-Dyck Paths = Cat ( a , b ). 9 / 47
Rational Christoffel words Catalan Numbers and Music Franck Definition Jedrzejewski The upper (lower) Christoffel path of slope b / a is the path from (0 , 0) to ( a , b ) in Catalan Numbers the integer lattice Z × Z that satisfies the following two conditions : (i) The path lies above (below) the line segment that begins at the origin and Rational Catalan Numbers ends at ( a , b ). Dyck Path (ii) The region in the plane enclosed by the path and the line segment contains no Christoffel words other points of Z × Z besides those of the path. Well-formed Scales Definition Narayana Numbers Christoffel path of slope b / a determines a word w in the alphabet { x , y } by Block Designs encoding steps of the first type by the letter x and steps of the second type by the Johson Works letter y . Catalan Designs Permutations Rational Associahedra 10 / 47
Rational Christoffel words Catalan Numbers and Music A note by C. Kassel. In Strasbourg, After French-Prussian War in 1870, France Franck lost Alsace-Lorraine to the German Empire. The Prussians created a new Jedrzejewski university in Strasbourg Christoffel founded the Mathematisches Institut in 1872. Catalan Numbers Rational Catalan Numbers Dyck Path Christoffel words Well-formed Scales Narayana Numbers Block Designs Johson Works Elwin Bruno Christoffel (1829–1900) Catalan Designs Permutations Observatio arithmetica, Annali di Matematica Pura ed Applicata , vol. 6 (1875), Rational Associahedra 148–152. Exemplum I. Sit a = 4, b = 11, erit series (r.) notis c , d ornata r. = 4 8 1 5 9 2 6 10 3 7 0 4 g. = c d c c d c c d c d c words as g=cdccdccdcdc are called Christoffel words 11 / 47
Rational Christoffel Duality Catalan Numbers and Music Franck Jedrzejewski Catalan Numbers Rational Catalan Numbers Dyck Path Christoffel words Well-formed Scales The scale : fa sol la si do ré mi fa ∼ { 5 , 7 , 9 , 11 , 0 , 2 , 4 , 5 } is encoded with a = tone , b = semi − tone Narayana Numbers the Christoffel word : aaabaab of slope 5/2 Block Designs Johson Works The same scale fa do sol ré la mi si (in the octave fa -fa ) Catalan Designs is encoded with x= fifth up, y = fourth down Permutations the dual Chirstoffel word xyxyxyy of slope 4/3 Rational Associahedra The dual Christoffel word w of slope a / b is the Christoffel word w ∗ of slope a ∗ / b ∗ with a ∗ and b ∗ are multiplicative inverse of a and b in Z / ( a + b ) Z . Example. The multiplicative inverse of 2 is 4 in Z 7 , and the inverse of 5 is 3, since 5 × 3 = 1 mod 7 and 2 × 4 = 1 mod 7 12 / 47
Rational Well-formed Scales Catalan Numbers and Music Franck Jedrzejewski Palindromic decomposition (See Kassel, Reteneuauer) Catalan • The lydian word aaabaab has a decompostion w = aub with u = aabaa Numbers palindromic Rational Catalan Numbers • And u has a decomposition u = rabs with r = a and s = aa palindromic. Dyck Path • The dual word w ∗ = xyxyxyy has the same decomposition Christoffel words Well-formed Scales The scale is well-formed (modulo 12) : 5-generated Narayana Numbers Block Designs 5 5 5 5 5 5 5 → 0 → 7 → 2 → 9 → 4 → 8 Johson Works Catalan Designs Permutations step = 3 : Rational Associahedra 5 7 9 11 0 2 4 5 7 9 11 0 2 4 5 7 9 11 0 2 4 5 7 9 11 0 2 4 5 13 / 47
Recommend
More recommend
