Unlabeled Motzkin numbers

Transcription

1 Dept. Computer Science and Engineering 013

2 Catalan numbers Catalan numbers can be defined by the explicit formula: C n = 1 n = n! n + 1 n n!n + 1! and the ordinary generating function: Cx = n=0 C n x n = 1 1 x. x Catalan numbers for n = 0, 1,... form the sequence A in OEIS: 1, 1,, 5, 1,, 13, 9, 130, 86, 16796, 58786,...

3 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: which can be further interpreted as the number of configurations of n noncrossing chords connecting n labeled points on a circle:

4 Motzkin numbers Catalan number C n represents the number of configurations of n noncrossing chords connecting 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: M n = n/ k=0 n C k. k The generating functions of Motzkin numbers is: Mx = M n x n = 1 x 1 x C 1 x = 1 x 1 x 3x x. n=0

5 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.

6 Cyclic and dihedral Motzkin numbers Cyclic Motzkin number Mn C 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 Mn C. Mn C = 1 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 Mn D 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 Mn D as follows: Mn D = 1 H c + H r = 1 n MC n + 1 H r. n c Cn r R n r R n

7 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 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 /. The number of special configurations of period n / equals M n/ 1. The number of special configurations of period n is Mn/ 1.

8 Chord configurations and periods Below we list of all configurations of chords connecting n n 6 vertices and specify their periods p. n = : p = 1 p = 1 n = 3: p = 1 p = 3 n = : p = 1 p = p = p = n = 5: p = 1 p = 5 p = 5 p = 5 p = 5 p = 1 p = p = 3 p = 3 p = 3 p = 3 n = 6: p = 6 p = 6 p = 6 p = 6 p = 6 p = 6

9 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.

10 Nonspecial configurations of fixed period Let bn, m be the number of nonspecial configurations on n vertices, whose period equals m. Clearly, bn, m can be non-zero only if m divides n. Rotation of a chord configuration of period m by the angle m 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 bn, m, k as the number of chord configurations on n vertices of period m with k n/m maximal chords. Similarly, let cn, m, k be the number of such configurations with a labeled maximal chord. Clearly, bn, m = k 0 bn, m, k.

11 Formula for bn, m, k Theorem For m n, m < n, and k 1, cn, m, k equals the coefficient of x m y k in Lemma x 1 ymx 1 x 1 = 1 y 1 x 1 1 x 3x. 1 x For m n and k 1, cn, m, k = bn, m /d, k /d k/d. d m,k Lemma For m n, we have bn, m, 0 = [m = 1] and for k 1, bn, m, k = 1 µd cn, k m /d, k /d, d m,k where µ is Möbius function.

12 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 ti. 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 + z + + t k + z k = m. Then cn, m, k as the total number of chords configurations within P equals t 1 +z 1 + +t k +z k =m M t1 M t M tk = [x m k ]Mx k 1 x k = [x m x k ] Mx. 1 x We multiply this by y k and sum over k 0 to get 1 cn, m, k = [x m y k ] 1 ymx x 1 x.

13 Generating function for bn, m We define the following functions: Bx = Fx = q=1 q=1 µq q ϕq q T x = ln 1 x x Mx, T x q = ln T x q = ln where ϕ is Euler totient function. Lemma 1 x q x q Mx q µq/q, q=1 1 x q x q Mx q ϕq/q, q=1 For positive integers m n with m < n, bn, m = [x m ] Bx.

14 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 if m divides n, M n = m n b n, m m, and Mn C = m n b n, m. Lemma For m n, we have b n, m = bn, m if m < n /. Furthermore, b n, n / = bn, n / + M n/ 1 if n is even, and b n, n = 1 n M n m n m<n b n, m m.

15 Cyclic Motzkin numbers Theorem The generating function for the number of asymmetric chord configurations b n, n is n=0 b n, n x n = 1 1 x Mx x Mx = 1 1 x Mx x Mx + lnmx + ln q + lnmx + Bx T x 1 x q x q Mx q µq/q. Theorem n=0 M C n x n = 1 1 x Mx + x Mx = 1 1 x Mx + x Mx + lnmx + ln q + lnmx + Fx T x 1 x q x q Mx q ϕq/q.

16 Dihedral Motzkin numbers: computing H r Lemma For an even n = m and a fixed reflection r R n about a diameter connecting centers of two arcs of the circle, we have Lemma H r = [x m Mx ] 1 x Mx. For an even n = m and a fixed reflection r R n about a diameter connecting two of the vertices, we have H r = [x m 1 ] 1 x Mx + Mm 1 = [x m 1 ] 1 x Mx + x Mx. Lemma For an odd n = m + 1 and a fixed reflection r R n, we have H r = [x m ] Mx 1 x Mx.

17 Dihedral Motzkin numbers Lemma For even n = m, we have 1 + Mx H r = [x m ] m 1 x Mx + x Mx = [x n ] n r R n For odd n = m + 1, Theorem For any integer n 0, 1 + Mx 1 x Mx + x Mx. H r = [x m Mx ] m x Mx = [x n x Mx ] n 1 x Mx. r R n 1 H r = [x n ] n r R n 1 + x + 1 Mx 1 x Mx + x Mx.

18 Dihedral Motzkin numbers Using formula Mn D = 1 MC n + 1 n r R n H r, we deduce the generating function for dihedral Motzkin numbers: n=0 M D n x n = 1 1 x Mx + x Mx lnmx + Fx T x x + 1 Mx 1 x Mx

19 Unlabeled Motzkin number in OEIS Currently there are three sequences in the Online Encyclopedia of Integer Sequences OEIS related to unlabeled Motzkin numbers: A17595 Cyclic Motzkin numbers A Number of asymmetric w.r.t. rotations chord configurations b n, n A Dihedral Motzkin numbers

20 Acknowledgements Dr. Glenn Tesler, University of California, San Diego National Science Foundation grant no. IIS-15361