0,6

A Motzkin 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=(1,0) and never going below the x-axis. Motzkin paths are counted by the Motzkin numbers (A001006).

A trapezoid in a Motzkin 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 Motzkin 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 Motzkin path is the sum of the heights of its maximal trapezoids. For example, in the Motzkin path w=UH(UHD)D(UUDD) we have two maximal trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3.

This concept is analogous to the concept of pyramid weight in a Dyck path (see the Denise-Simion paper).

Row sums yield the Motzkin numbers (A001006).

Row n has 1+floor(n/2) terms.

T(2n+1,n)=(n+2)*2^(n-1) (A001792).

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

G.f.=G=G(t, z) satisfies G=1+zG+z^2[G-(1-t)/((1-z)(1-tz^2))]G.

Triangle begins:

1;

1;

1,1;

1,3;

1,6,2;

1,12,8;

1,24,22,4;

T(4,0)=1,T(4,1)=6, T(4,2)=2 because the nine Motzkin paths of length 4, namely HHHH, HH(UD),H(UD)H,H(UHD),(UD)HH,(UD)(UD),(UHD)H,(UHHD),(UUDD), have trapezoid weights 0,1,1,1,1,2,1,1,2, respectively; the maximal trapezoids are shown between parentheses.

Cf. A001006, A001792, A104574.

Adjacent sequences: A104570 A104571 A104572 this_sequence A104574 A104575 A104576

Sequence in context: A130302 A134546 A055151 this_sequence A010467 A122913 A069115

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 16 2005