中央研究院 数學研咒所 Theorem: Let S=((, 1 )U(( -i+1)i=1,2 W(O)=(-+1)=an Let L,= l (s) be the set of s-nonnegativ e paths with endpoint(n,0 andL=L(S)=∪, Define a generating function=∑∑w(P) ≥0P∈L Then 件()=1+∑∑a2f() 1=1 J 1+f(=)°·F(=,f(=) 第11页
第11页 1 ( ) ( , ( )). Then 1 [ ( )] and ( ) .Define a generating function ( ) . Let ( ) be the set of S- nonnegativ e paths with endpoint (n,0) (01) 1 ( 1) . Theorem :Let {(0,1)} {( , 1) 1,2, , 1,2, , )} 1 1 0 0 zf z F z f z f(z) a z f z L L S L f(z) w P z L L S w , ,w j,-i a S j i i r j m j i r i m j i j n P L n n n n n i j n = + • = + = = = = = + = = − + = = = =
中央研究院 数學研咒所 a decomposition of a s-nonnegative path P=10,DP10,DP21O,1P3.P1(,计+1P (0,1)=1,n(,++1)= [f(z) 第12页
第12页 • 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
中央研究院 数學研咒所 More general cases Let n be a function from Zo Xz to Zso For any lattice path P=(x,y1(2, 22).k,,k) define the n-length of P, denoted by n(p),as k (P)=∑4(x2y) 第13页
第13页 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 = = k i i i P x y 1 ( ) ( , )
中央研究院 数學研咒所 Theorem: Let S=B1∪(a+1)=12,…r,j=12,,m)and(00)S, where a and B, are nonnegative e integers. Let 1)(.)+(an1+1)=,B1)≠.andw()w(xn+1)=an Let l, =Ln (s)be the set of s-nonnegativ e paths with n-length n ending at a point on x-axis and L=L(S)=UL,(S) n≥0 Define a generating function f(3)=w(P)=PThen 件6)=1+2∑a1=1D17 1+f(-)F(=2,f(=) 第14页
第14页 1 ( ) ( , ( )). 1 Define a generating function .Then a point on - axis and . Let ( ) be the set of S- nonnegativ e paths with -length ending at ( 1) ( ,1) ( 1) ( ,1) 0,and ( ,1) 1 . where and are nonnegativ e integers . Let {( ,1)} {( , 1) 1,2, , 1,2, , )} and (0,0) , Theorem : Let 1 1 0 1 zf z F z f z f(z) a z [f(z)] f(z) w(P)z x L L(S) L (S) L L S n i- ,-i j, w w w ( ,-i ) a S i i r j m S j i r i m j i j P L λ(P) n n n n j i j i j j j = + = + = = = = + + = + = = − + = = = = −
中央研究院 数學研咒所 Uniform partition An n-Dyck path is a lattice path from(0, 0)to(2n, 0) in the plane integer lattice ZXZ consisting of up-step(l, 1)and down-step(,-1) ◆◆ 2n The number of n-Dyck paths is n 第15页
第15页 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). The number of n-Dyck paths is n 2n