Combinatorial interpretations for a class of algebraic equations and uniform partitions

1. Combinatorial interpretations for a class of algebraic equations and uniform partitions Speaker: Yeong-Nan Yeh Institute of mathemetics, Academia sinica Aug. 21, 2012

2. Catalan paths An n-Catalan path is a lattice path from (0,0) to (2n,0) in the first quadrant consisting of up-step (1,1) and down-step (1,-1) . 第2页

3. Catanlan number 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, … , 第3页

4. 第4页

5. Motzkin paths An n-Motizkin path is a lattice path from (0,0) to (n,0) in the first quadrant consisting of up-step (1,1), level-step (1,0) and down-step (1,-1). Motzkin number:1, 1, 2, 4, 9, 21, 51, 127, 323, 835, … , 第5页

6. 第6页

7. Given an algebaric equation for arbitrary polynomial F(z,y) , how to construct a combinatorial structure such that its generating function f(z) satisfies this equation? 第7页

8. Lattice paths • A lattice path is a sequence (x1,y1)(x2,y2)…(xk,yk) of vectors in the plane with (xi,yi)∈Z≥0×Z\{(0,0)}, where Z and Z≥0 are the sets of integers and nonnegative integers respectively.

9. Weight of a lattice path • Let w be a function from Z≥0×Z to R, where R is the set of real numbers. • For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the weight of P, denoted by w(P), as

10. S-path and S-nonnegative path • Let S be a finite subset of Z≥0×Z\{(0,0)}. • An S-path is a lattice path (x1,y1)(x2,y2)…(xk,yk) with (xi,yi)∈S. • An S-nonnegative path is an S-path in the first quadrant

11. 第11页

12. A decomposition of a S-nonnegative path. P=(0,1)P1(0,1)P2(0,1)P3…Pi-1(j,-i+1)Pi w(0,1)=1,w(j,-i+1)=ai,j 第12页

13. More general cases • Let λ be a function from Z≥0×Z to Z≥0. • For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the λ-length of P, denoted by λ(P), as

14. 第14页

15. The number of n-Dyck paths is Uniform partition • An n-Dyck path is a lattice path from (0,0) to (2n,0) in the plane integer lattice Z×Z consisting of up-step (1,1) and down-step (1,-1).

16. Chung-Feller theorem: • The number of Dyck path of semi-length n with m up-steps under x-axis is the n-th Catalan number and independent on m. K.L. Chung, W. Feller, On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) 605-608 第16页

17. Uniform partition (An uniform partition for Dyck paths) The number of up-steps (1,1) lying below x-axis 第17页

19. Lifted Motzkin paths A lifted n-Motizkin path is a lattice path from (0,0) to (n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1), which never passes below the line y=1 except (0,0). 第19页

20. Free Lifted Motzkin paths A freelifted n-Motizkin path is a lattice path from (0,0) to (n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1). 第20页

21. 第21页 The number of free lifted n-Motzkin path with m steps at the left of the rightmost lowest point is the n-th Motzkin number and independent on m.

22. An uniform partition for free lifted Motzkin paths Shapiro found an uniform partition for Motzkin path.L. Shapiro, Some open questions about random walks, involutions, limiting distributions, and generating functions, Advances in Applied Math. 27 (2001), 585-596. The number of steps at theleft of the rightmost lowest point of a lattice path • Eu, Liu andYeh proved this proposition.Eu, S. P. Liu, S. C. and Yeh, Y. N. Taylor expansions for Catalan and Motzkin numbers, Adv. Appl. Math. 29 (2002) 345-357 第22页

24. Function of uniform partition type • For any generating function f(x),the form is called the function of uniform partition type for f(x).

25. Function of uniform partition type 第25页

26. 第26页

27. 第27页

28. A decomposition of a S-nonnegative path. (j-i+1,-i+1) P=(1,1)P1(1,1)P2(1,1)P3…Pi-1(j-i+1,-i+1)Pi w(1,1)=1, w(j-i+1,-i+1)=ai,j 第28页

29. Function of uniform partition type

30. Combinatorial interpretations for H(y,z) and G(y,z)? 第30页

31. Combinatorial interpretation for H(y,z) • Recall that an S-path is a lattice path P=(x1,y1)(x2,y2)…(xk,yk) with (xi,yi)∈S. • Define the nonpositive length of P, denoted by nl(P), as the sum of x-coordinate of steps touching or going below x-axis. 第31页

32. Combinatorial interpretation for H(y,z) • Define the nonpositive length of P, denoted by nl(P), as the sum of x-coordinate of steps touching or going below x-axis, nl(P)=1+1+1+2+1+1=7 第32页

33. Combinatorial interpretation for H(y,z) 第33页

34. f(z) f(z) f(z) H(y,z) f(yz) f(yz) f(yz) • A decomposition of a S-path. 第34页

35. Combinatorial interpretation for G(y,z) • A rooted S-nonnegative path is a pair [P;k] consisting of an S-nonnegative path P=(x1,y1)(x2,y2)…(xn,yn) with xn≥1 and a nonnegative integer k with 0≤ k≤ xn-1. • For example, P=(1,1)(1,1)(1,-2)(1,0)(1,1)(1,1)(1,1)(1,-1)(2,-1).[P;0],[P;1] and [P;2] are rooted S-nonnegative path.

36. Combinatorial interpretation for G(y,z) 第36页

37. f(z) f(z) f(z) f(z) • A decomposition of a rooted S-nonnegative path. k 第37页

38. Combinatorial interpretation for CF(y,z) • A lifted S-path is an S-path in the plane starting at (0,0) and ending at a point in the line y=1. • A rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x1,y1)(x2,y2)…(xn,yn) with xn≥1 and a nonnegative integer k with 0≤ k≤ xn-1.

39. Combinatorial interpretation for CF(y,z)

40. A decomposition for an rooted lifted S-path G(y,z) H(y,z) (n+1,1) (0,0) (n+1-k,0) The last step (1,1) from y=0 to y=1 第40页

41. More general cases • Let λ be a function from Z≥0×Z to Z≥0. • For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the λ-length of P, denoted by λ(P), as

42. A λ-rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x1,y1)(x2,y2)…(xn,yn) with λ(xn,yn)≥1 and a nonnegative integer k with 0≤ k≤ λ(xn,yn)-1.

43. For any S-path P=(x1,y1)(x2,y2)…(xn,yn), define the λ-nonpositive length of P, denoted by nlλ(P), as the sum of λ-length of steps touching or going below x-axis. • For example, let λ(1,1)=0, λ(x,y)=x for any(x,y)≠（1,1） nlλ(P)=1+1+0+2+1+0=5 • For any rooted lifted S-path [P;k], define the rootedλ-nonpositive length of P as nlλ(P)+k.