1,2

A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). A trapezoid in a Schroeder path is a factor of the form U^i H^j D^i (i>=1, j>=0), i being the height of the trapezoid. A trapezoid in a Schroeder path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The trapezoid weight of a Schroeder path is the sum of the heights of its maximal trapezoids. For example, in the Schroeder path w=UH(UHD)D(UUDD) we have two trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3. This concept is an analogous to the concept of pyramid weight in a Dyck path (see the Denise-Simion paper). Partial sums of A047665 which, in turn, are the partial sums of A002002.

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).

G.f.=[1-z-sqrt(1-6z+z^2)]/[2(1-z)^2*sqrt(1-6z+z^2)]

a(2)=7 because the six Schroeder paths of length 4, namely HH,(UD)H,H(UD),(UHD), (UD)(UD) and (UUDD), have trapezoid weights 0,1,1,1,2 and 2, respectively; the maximal trapezoids are shown between parentheses.

G:=(1-z-sqrt(1-6*z+z^2))/2/(1-z)^2/sqrt(1-6*z+z^2):Gser:=series(G, z=0, 28): seq(coeff(Gser, z^n), n=1..25);

Cf. A006318, A047665, A002002, A104552.

Sequence in context: A141845 A048437 A099461 this_sequence A027241 A056197 A158576

Adjacent sequences: A104550 A104551 A104552 this_sequence A104554 A104555 A104556

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2005