A182900 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k valleys. The members of B(n) are paths of weight n that start at (0,0), end on but never go below 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. A valley is a (1,-1)-step followed by a (1,1)-step.
1, 1, 2, 4, 8, 17, 36, 1, 78, 4, 171, 14, 379, 43, 1, 848, 125, 5, 1912, 351, 20, 4341, 960, 71, 1, 9915, 2579, 235, 6, 22767, 6833, 745, 27, 52526, 17916, 2281, 108, 1, 121698, 46593, 6805, 399, 7, 283043, 120385, 19885, 1400, 35, 660579, 309416, 57141, 4712, 155, 1
Offset: 0
Examples
T(7,1)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDUDh, UDUhD, and UhDUD. Triangle starts: 1; 1; 2; 4; 8; 17; 36,1; 78,4; 171,14;
References
- M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Programs
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Maple
eq := z^3*(z+z^2+v-v*z-v*z^2)*F^2-(1-z-z^2-z^3+v*z^3)*F+1 = 0: F := RootOf(eq, F): Fser := simplify(series(F, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Fser, z, n)) end do: 1; 1; 2; for n from 0 to 18 do seq(coeff(P[n], v, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form
Formula
G. f.: F=F(v,z) satisfies z^3*(z+z^2+v-vz-vz^2)F^2 - (1-z-z^2-z^3+vz^3)F+1=0 (z marks weight, v marks number of valleys).
The trivariate g.f. H(u,v,z), where u (v) marks peaks (valleys) and z marks weight is given by H=1+zH+z^2*H+z^3*(u-1+H)[v(H-1-zH-z^2*H)+1+zH+z^2*H].
Comments