OFFSET
0,3
COMMENTS
REFERENCES
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
Let F=F(t,s,x,y,z) be the 5-variate g.f. of the considered weighted lattice paths, where z marks weight, t (s) marks number of peaks (valleys), x (y) indicates that the path starts with a (1,1)-step ((1,-1)-step). Then F(t,s,x,y,z)=1+z(1+z)F(t,s,1,1,z)+xz^3[t+H(t,s,z)-1]F(t,s,s,1,z)+yz^3[s+H(s,t,z)-1]F(t,s,1,t,z), where H=H(t,s,z) is given by H=1+zH+z^2*H+z^3*(t-1+H)[s(H-1-zH-z^2*H)+1+zH+z^2*H] (see A182900).
EXAMPLE
T(7,2)=3. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDhUD, UDUDh.
Triangle starts:
1;
1;
2;
4,1;
9,2;
21,5;
48,14,1;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 16 2010
STATUS
approved