OFFSET
0,15
COMMENTS
A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
T(n,n)=A000045(n-1) (n>=1; the Fibonacci numbers).
Sum(k*T(n,k), k>=0)=A166302(n).
LINKS
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
FORMULA
G.f.: G=G(t,z) satisfies G = 1 + zG[G - 1 + tz - tz(1 - t)/(1 - tz)].
EXAMPLE
T(6,5)=2 because we have U(UUDD)(UUUDDD)D and U(UUUDDD)(UUDD)D (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
0,0;
0,0,1;
0,0,0,1;
0,0,0,0,2;
0,0,0,0,1,3;
0,0,0,0,1,2,5;
0,0,0,0,1,2,6,8;
MAPLE
eq := G = 1+z*G*(G-1+t^2*z*(1-z)/(1-t*z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 07 2009
STATUS
approved