A182898 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k returns to the horizontal axis (both from above and below). The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 46, 4, 21, 112, 20, 34, 262, 80, 55, 600, 268, 8, 89, 1356, 816, 56, 144, 3046, 2324, 280, 233, 6832, 6320, 1144, 16, 377, 15354, 16620, 4136, 144, 610, 34658, 42652, 13728, 864, 987, 78706, 107520, 42816, 4144, 32, 1597, 180000, 267564, 127392, 17264, 352
Offset: 0
Examples
T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely ud and du, have 1 return to the horizontal axis. Triangle starts: 1; 1; 2; 3,2; 5,6; 8,18; 13,46,4; 21,112,20;
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.
Programs
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Maple
eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-2*t*z^3*c): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
Formula
G.f.: G(t,z) =1/[1-z-z^2-2tz^3*c], where c satisfies c = 1+zc+z^2*c+z^3*c^2.
The trivariate g.f. H=H(t,s,z), where t (s) marks (1,-1)-returns ((1,1)-returns) to the horizontal axis, and z marks weight is given by H=1+zH+z^2*H+(t+s)z^3*cH, where c satisfies c = 1+zc+z^2*c+z^3*c^2.
Comments