Unlabeled Motzkin numbers Max Alekseyev Dept. Computer Science and - PowerPoint PPT Presentation

Unlabeled Motzkin numbers Max Alekseyev Dept. Computer Science and Engineering 2013 Max Alekseyev Unlabeled Motzkin numbers

Catalan numbers Catalan numbers can be defined by the explicit formula: � 2 n � 1 (2 n )! C n = = n + 1 n !( n + 1)! n and the ordinary generating function: C n · x n = 1 − √ 1 − 4 x ∞ � C ( x ) = . 2 x n =0 Catalan numbers for n = 0 , 1 , . . . form the sequence (A000108 in OEIS): 1 , 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , 16796 , 58786 , . . . Max Alekseyev Unlabeled Motzkin numbers

Chord configurations We are particularly interested in the combinatorial interpretation of C n as the number of expressions containing n properly embedded pairs of parentheses. For n = 3, these expressions are: ( ( ) ( ) ) ( ( ( ) ) ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( ) ) ( ) 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 which can be further interpreted as the number of configurations of n noncrossing chords connecting 2 n labeled points on a circle: 2 2 2 2 2 1 1 1 1 1 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 5 5 5 5 5 Max Alekseyev Unlabeled Motzkin numbers

Motzkin numbers Catalan number C n represents the number of configurations of n noncrossing chords connecting 2 n labeled points on a circle. Motzkin number M n represents the number of configurations of (any number of) noncrossing chords connecting n labeled points on a circle. We can easily expressed Motzkin numbers in terms of Catalan numbers: � n ⌊ n / 2 ⌋ � � M n = C k . 2 k k =0 The generating functions of Motzkin numbers is: √ ∞ x 2 1 − 2 x − 3 x 2 1 � � = 1 − x − M n · x n = � M ( x ) = 1 − x · C . (1 − x ) 2 2 x 2 n =0 Max Alekseyev Unlabeled Motzkin numbers

Unlabeled Motzkin numbers Consider a circle with n equally spaced points, which we will call vertices . A set of noncrossing chords connecting vertices is called a chord configuration . We define two types of unlabeled Motzkin numbers counting the number of chord configurations on unlabeled vertices. Namely, we define cyclic and dihedral Motzkin numbers counting the number of chord configurations up to the action of cyclic and dihedral groups, respectively. Max Alekseyev Unlabeled Motzkin numbers

Cyclic and dihedral Motzkin numbers Cyclic Motzkin number M C n represents the number of chord configurations on n vertices under the action of the cyclic group (of rotations ) C n . Burnside lemma allows us to give the following expression for M C n . n = 1 � M C H c , (1) n c ∈ C n where H c is the number of chord configurations invariant w.r.t. c . Similarly, dihedral Motzkin number M D n represents the number of chord configurations on n vertices under the action of the dihedral group D n . Viewing elements of D n as n rotations, forming the cyclic subgroup C n , and n reflections , forming a set R n , we compute M D n as follows:   n = 1  = 1 n + 1 H c + M D  � � H r 2 M C � H r . (2) 2 n 2 n c ∈ C n r ∈ R n r ∈ R n Max Alekseyev Unlabeled Motzkin numbers

Periods and special configurations We define the period of a chord configuration S as the smallest positive integer p such that S is invariant w.r.t. rotation of the circle by the angle p · 2 π n . Clearly, the period of any chord configuration on n vertices divides n . A chord configuration on n vertices is called special if it contains a chord connecting two diametrically opposite vertices. Special configurations exist only for even n . Period of a special configuration can be only n or n / 2 . The number of special configurations of period n / 2 equals M n / 2 − 1 . � M n / 2 − 1 � The number of special configurations of period n is . 2 Max Alekseyev Unlabeled Motzkin numbers

Chord configurations and periods Below we list of all configurations of chords connecting n (2 ≤ n ≤ 6) vertices and specify their periods p . n = 2: p = 1 p = 1 n = 3: p = 1 p = 3 n = 4: p = 1 p = 2 p = 2 p = 4 p = 1 p = 5 p = 5 p = 5 p = 5 n = 5: p = 1 p = 2 p = 3 p = 3 p = 3 p = 3 n = 6: p = 6 p = 6 p = 6 p = 6 p = 6 p = 6 Max Alekseyev Unlabeled Motzkin numbers

Nonspecial chord configurations A chord c in a nonspecial chord configuration partition the vertices other than the endpoints of c into two subsets formed by vertices laying at the same side of c . We define the span of c as the smaller of these subsets together with the endpoints of c . For a configuration of period m < n , the span of each chord does not exceed m . A chord is called maximal if its endpoints do not reside within the span of any other chord. It is easy to see that all chords in a chord configuration reside within the spans of the maximal chords. Max Alekseyev Unlabeled Motzkin numbers

Nonspecial configurations of fixed period Let b ( n , m ) be the number of nonspecial configurations on n vertices, whose period equals m . Clearly, b ( n , m ) can be non-zero only if m divides n . Rotation of a chord configuration of period m by the angle m · 2 π n translates maximal chords into maximal chords. Therefore, the number of maximal chords in such configuration must be a multiple of n / m . For m | n , define b ( n , m , k ) as the number of chord configurations on n vertices of period m with k · n / m maximal chords. Similarly, let c ( n , m , k ) be the number of such configurations with a labeled maximal chord. Clearly, � b ( n , m ) = b ( n , m , k ) . k ≥ 0 Max Alekseyev Unlabeled Motzkin numbers So our goal is to find b ( n , m , k ).

Formula for b ( n , m , k ) Theorem For m | n, m < n, and k ≥ 1 , c ( n , m , k ) equals the coefficient of x m y k in √ � − 1 � − 1 x 2 � � 1 − y 1 − x − 1 − 2 x − 3 x 2 1 − y M ( x ) = . 1 − x 2(1 − x ) Lemma For m | n and k ≥ 1 , � c ( n , m , k ) = b ( n , m / d , k / d ) · k / d . d | ( m , k ) Lemma For m | n, we have b ( n , m , 0) = [ m = 1] and for k ≥ 1 , b ( n , m , k ) = 1 � µ ( d ) · c ( n , m / d , k / d ) , k d | ( m , k ) where µ ( · ) is M¨ obius function. Max Alekseyev Unlabeled Motzkin numbers

Proof of theorem Proof. Consider an arbitrary chord configuration on n vertices with period dividing m and containing k · n / m maximal chords, one of which is labeled. Let P be set of m consecutive vertices on the circle that starts with the counterclockwise endpoint of the labeled maximal chord and goes clockwise. Then P contains the spans of k maximal chords. Let t i (1 ≤ i ≤ k ) be the size of the span of the i -th (counting clockwise) maximal chord in P . Then the number of chord configurations within this span is M t i − 2 . Let z i (1 ≤ i ≤ k ) be the number of vertices between the endpoints of i -th and ( i + 1)-th maximal chords (or the end of P for i = k ) so that the total number of vertices is t 1 + z 1 + t 2 + z 2 + · · · + t k + z k = m . Then c ( n , m , k ) as the total number of chords configurations within P equals � k x 2 � M t 1 − 2 · M t 2 − 2 · · · M t k − 2 = [ x m − 2 k ] M ( x ) k (1 − x ) − k = [ x m ] � M ( x ) . 1 − x t 1 + z 1 + ··· + t k + z k = m We multiply this by y k and sum over k ≥ 0 to get � − 1 � 1 − y M ( x ) x 2 c ( n , m , k ) = [ x m y k ] . 1 − x Max Alekseyev Unlabeled Motzkin numbers

Generating function for b ( n , m ) We define the following functions: 1 − x − x 2 · M ( x ) � � T ( x ) = − ln , ∞ ∞ µ ( q ) � − µ ( q ) / q , 1 − x q − x 2 q · M ( x q ) � � · T ( x q ) = ln � B ( x ) = q q =1 q =1 ∞ ∞ ϕ ( q ) � − ϕ ( q ) / q , 1 − x q − x 2 q · M ( x q ) � · T ( x q ) = ln � � F ( x ) = q q =1 q =1 where ϕ ( · ) is Euler totient function. Lemma For positive integers m | n with m < n, b ( n , m ) = [ x m ] B ( x ) . Max Alekseyev Unlabeled Motzkin numbers

Configurations of fixed period Let b ′ ( n , m ) be the number of chord configurations (both special and nonspecial) whose period equals m . Clearly, b ′ ( n , m ) can be non-zero only m | n b ′ ( n , m ) · m , and M C m | n b ′ ( n , m ). if m divides n , M n = � n = � Lemma For m | n, we have b ′ ( n , m ) = b ( n , m ) if m < n / 2 . Furthermore, b ′ ( n , n / 2 ) = b ( n , n / 2 ) + M n / 2 − 1 if n is even, and   b ′ ( n , n ) = 1 � b ′ ( n , m ) · m  M n −  .   n m | n m < n Max Alekseyev Unlabeled Motzkin numbers

Cyclic Motzkin numbers Theorem The generating function for the number of asymmetric chord configurations b ′ ( n , n ) is ∞ b ′ ( n , n ) · x n = 1 − (1 − x ) · M ( x ) − x 2 · M ( x 2 ) � + ln( M ( x )) + B ( x ) − T ( x ) 2 n =0 = 1 − (1 − x ) · M ( x ) − x 2 · M ( x 2 ) � − µ ( q ) / q � 1 − x q − x 2 q · M ( x q ) � + ln( M ( x )) + ln . 2 q ≥ 2 Theorem ∞ n · x n = 1 − (1 − x ) · M ( x ) + x 2 · M ( x 2 ) � M C + ln( M ( x )) + F ( x ) − T ( x ) 2 n =0 = 1 − (1 − x ) · M ( x ) + x 2 · M ( x 2 ) � − ϕ ( q ) / q 1 − x q − x 2 q · M ( x q ) � � + ln( M ( x )) + ln . 2 q ≥ 2 Max Alekseyev Unlabeled Motzkin numbers