OFFSET
0,5
COMMENTS
A trapezoid in a peakless Motzkin path is a factor of the form U^i H^j D^i (i, j>=1), i being the height of the trapezoid and U=(1,1), H=(1,0), D=(1,-1). A trapezoid in a peakless 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 peakless Motzkin path is the sum of the heights of its maximal trapezoids. For example, in the peakless Motzkin path w=UH(UHD)D(UUHHDD) 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).
LINKS
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
FORMULA
a(n) = Sum(k*A171852(n,k), k>=0).
G.f.=z^3*g/[(1 + z)(1 - z + z^2 - 2z^2*g)(1 - z)^2], where g=g(z) satisfies g=1+zg+z^2*g(g-1).
Conjecture: (-n+1)*a(n) +(3*n-5)*a(n-1) +(-n+7)*a(n-2) -7*a(n-3) -5*a(n-4) -7*a(n-5) +n*a(n-6) +(-3*n+16)*a(n-7) +(n-6)*a(n-8)=0. - R. J. Mathar, Dec 07 2017
EXAMPLE
a(4)=3 because the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely HHHH, HUHD, UHHD and UHDH have trapezoid weights 0, 1, 1 and 1, respectively.
MAPLE
eqg := g = 1+z*g+z^2*g*(g-1): g := RootOf(eqg, g): G := z^3*g/((1+z)*(1-z)^2*(1-z+z^2-2*z^2*g)): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 35);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 08 2010
STATUS
approved